A.A. 2023/2024
A.A. 2023/2024
Giovedì 4 Aprile 2024, ore 15.30, Aula 207
Giovedì 4 Aprile 2024, ore 15.30, Aula 207
Mark L. Lewis (Kent State University, USA)
Mark L. Lewis (Kent State University, USA)
Title: A lower bound on the size of maximal abelian subgroups
Title: A lower bound on the size of maximal abelian subgroups
Abstract: Let G be a p-group for some prime p. Let n be the positive integer so that |G:Z(G)|=p^n. Suppose A is a maximal abelian subgroup of G. Let p^l=max\{|Z(C_G(g)):Z(G)| : g in G\Z(G)}, let p^b=max\{|cl(g)| : g in G\Z(G)}, and let p^a=|A:Z(G)|. Then we show that a is at least n/(b+l).
Abstract: Let G be a p-group for some prime p. Let n be the positive integer so that |G:Z(G)|=p^n. Suppose A is a maximal abelian subgroup of G. Let p^l=max\{|Z(C_G(g)):Z(G)| : g in G\Z(G)}, let p^b=max\{|cl(g)| : g in G\Z(G)}, and let p^a=|A:Z(G)|. Then we show that a is at least n/(b+l).
Mercoledì 13 Marzo 2024, ore 15.30, Aula 103
Mercoledì 13 Marzo 2024, ore 15.30, Aula 103
Title: Spin representations of symmetric groups in characteristic 2
Title: Spin representations of symmetric groups in characteristic 2
Abstract: Let G be a finite group and p a prime. Then there is a well-defined (at the level of composition factors) process of p-modular reduction for representations of G. It sometimes happens that two different irreducible modules in characteristic 0 can become the same when reduced modulo p, and it is interesting to determine exactly when this happens. For example, if G is the symmetric group, and two ordinary irreducibles are obtained from each other by tensoring with the sign representation, then their reductions modulo 2 will be the same.
Abstract: Let G be a finite group and p a prime. Then there is a well-defined (at the level of composition factors) process of p-modular reduction for representations of G. It sometimes happens that two different irreducible modules in characteristic 0 can become the same when reduced modulo p, and it is interesting to determine exactly when this happens. For example, if G is the symmetric group, and two ordinary irreducibles are obtained from each other by tensoring with the sign representation, then their reductions modulo 2 will be the same.
In this talk we consider this problem for the double covers of the symmetric groups in characteristic 2; in fact, we solve the more general problem of when the 2-modular reductions of two modules are proportional to each other. I will give the result, and explain some of the techniques used to prove it.
In this talk we consider this problem for the double covers of the symmetric groups in characteristic 2; in fact, we solve the more general problem of when the 2-modular reductions of two modules are proportional to each other. I will give the result, and explain some of the techniques used to prove it.
(joint with Eoghan McDowell)
(joint with Eoghan McDowell)
Mercoledì 13 Marzo 2024, ore 14.30, Aula 202
Mercoledì 13 Marzo 2024, ore 14.30, Aula 202
Rachel D. Camina (University of Cambridge, UK)
Rachel D. Camina (University of Cambridge, UK)
Title: Covering of groups
Title: Covering of groups
Abstract: Suppose G is a group, a covering of G is a set of proper subgroups whose union is G. The study of coverings of groups has a long history. In 2017, I was asked a question about coverings of finite p-groups. We now have an answer. I will talk about how we reached this answer and how we were led to the infinite world of pro-p groups.
Abstract: Suppose G is a group, a covering of G is a set of proper subgroups whose union is G. The study of coverings of groups has a long history. In 2017, I was asked a question about coverings of finite p-groups. We now have an answer. I will talk about how we reached this answer and how we were led to the infinite world of pro-p groups.
Giovedì 1 Febbraio 2024, ore 15.30, Aula Tricerri
Giovedì 1 Febbraio 2024, ore 15.30, Aula Tricerri
Title: Brauer's Problem 21
Title: Brauer's Problem 21
Abstract: In 1963, Brauer published a list of 43 problems that he saw as the main questions to be answered in group theory and character theory. This list has lead the research in the field of representation theory of finite groups since then. Many important problems in our area are collected in this list, such as the very recently solved Brauer's Height Zero Conjecture (Problem 23).
Abstract: In 1963, Brauer published a list of 43 problems that he saw as the main questions to be answered in group theory and character theory. This list has lead the research in the field of representation theory of finite groups since then. Many important problems in our area are collected in this list, such as the very recently solved Brauer's Height Zero Conjecture (Problem 23).
Problem 21 of Brauer's collection asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. In this talk we will discuss some aspects of this problem. We will specifically present its resolution for principal blocks in a joint work with A. Moretó and M. Schaeffer Fry.
Mercoledì 23 Gennaio 2024, ore 14:30, Aula Tricerri
Mercoledì 23 Gennaio 2024, ore 14:30, Aula Tricerri
Title: On pyramidal groups of prime power degree
Title: On pyramidal groups of prime power degree
Abstract: A Kirkman Triple System (KTS) is called m-pyramidal if there exists a subgroup G of its automorphism group that fixes m points of the KTS and acts regularly on the other points. Such a group G admits a unique conjugacy class C of involutions (elements of order 2) and |C|=m. We call groups with this property m-pyramidal. We prove that, if m is an odd prime power p^k, then every m-pyramidal group is solvable if and only if either m=9 or k is odd. We also determine the sizes of the vertex sets of the m-pyramidal KTS when m is a prime number. This is a joint work with X. Gao.
Abstract: A Kirkman Triple System (KTS) is called m-pyramidal if there exists a subgroup G of its automorphism group that fixes m points of the KTS and acts regularly on the other points. Such a group G admits a unique conjugacy class C of involutions (elements of order 2) and |C|=m. We call groups with this property m-pyramidal. We prove that, if m is an odd prime power p^k, then every m-pyramidal group is solvable if and only if either m=9 or k is odd. We also determine the sizes of the vertex sets of the m-pyramidal KTS when m is a prime number. This is a joint work with X. Gao.
References
References
X. Gao and M. Garonzi. On pyramidal groups whose number of involutions is a prime power. https://arxiv.org/pdf/2311.16690.pdf
X. Gao and M. Garonzi. On pyramidal groups whose number of involutions is a prime power. https://arxiv.org/pdf/2311.16690.pdf
Giovedì 30 Novembre 2023, ore 15.00, Aula Tricerri
Giovedì 30 Novembre 2023, ore 15.00, Aula Tricerri
Marco Fusari (Università degli Studi di Milano-Bicocca)
Marco Fusari (Università degli Studi di Milano-Bicocca)
Title: Cliques in derangement graphs for innately transitive groups
Title: Cliques in derangement graphs for innately transitive groups
Abstract: Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$.
Abstract: Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$.
Mercoledì 22 Novembre 2023, ore 15.30, Aula Tricerri
Mercoledì 22 Novembre 2023, ore 15.30, Aula Tricerri
Title: The Modular Isomorphism Problem for 2-generated abelian groups with cyclic commutator
Title: The Modular Isomorphism Problem for 2-generated abelian groups with cyclic commutator
Abstract: The Modular Isomorphism Problem asks whether the isomorphism type of the group algebra FG of a finite p-group over the field with p elements determines the isomorphism type of G.
Abstract: The Modular Isomorphism Problem asks whether the isomorphism type of the group algebra FG of a finite p-group over the field with p elements determines the isomorphism type of G.
We will present some results on this problem for the particular case where G is two-generated and G' is cyclic. This includes the unique known counter-example to the Modular Isomorphism Problem and some positive results for the case where p is odd.
Mercoledì 11 Ottobre 2023, ore 15.30, Aula Tricerri
Mercoledì 11 Ottobre 2023, ore 15.30, Aula Tricerri
Title: Degree divisibility in character correspondences
Title: Degree divisibility in character correspondences
Abstract: We discuss situations in which it is possible to find character correspondences for the McKay and Alperin—McKay conjectures with good properties. In particular, we explore how these correspondences can interact with character degrees. This is joint work with Damiano Rossi.
Abstract: We discuss situations in which it is possible to find character correspondences for the McKay and Alperin—McKay conjectures with good properties. In particular, we explore how these correspondences can interact with character degrees. This is joint work with Damiano Rossi.
A.A. 2022/2023
A.A. 2022/2023
Martedì 13 Giugno 2023, ore 15.00, Aula Tricerri.
Martedì 13 Giugno 2023, ore 15.00, Aula Tricerri.
Title: Units in group rings and representation theory of groups of prime order
Title: Units in group rings and representation theory of groups of prime order
Mercoledì 31 Maggio 2023, ore 14.30, Aula Tricerri.
Mercoledì 31 Maggio 2023, ore 14.30, Aula Tricerri.
Title: Synchronizing automata via a primitive matrix set approach
Title: Synchronizing automata via a primitive matrix set approach
Abstract: Automata were introduced in computer science as models of computing devices: they are labeled multi-digraphs where the vertices represent the states of the device and the labels represents the different control commands. An automaton is synchronizing if there exists a sequence of commands (a reset word) that brings the automaton into a particular state regardless of the initial one.
Abstract: Automata were introduced in computer science as models of computing devices: they are labeled multi-digraphs where the vertices represent the states of the device and the labels represents the different control commands. An automaton is synchronizing if there exists a sequence of commands (a reset word) that brings the automaton into a particular state regardless of the initial one.
The Černy's conjecture, a long-standing open problem that has brought the attention of different communities among mathematicians and theoretical computer scientists, states that every synchronizing automaton on n vertices has a reset word of length at most (n-1)^2. In practice, few automata with such long reset words are known, also called slowly synchronizing automata, while the conjecture has been proved only for some families of automata.
In this talk we approach the synchronization process by making use of the notion of primitive matrix set, conceived as an extension of the concept of primitive matrix; synchronization is thus connected to the property of a labeled directed network of admitting a sequence of labels that links any pair of vertices. Primitive sets also find applications in stochastic switching systems, in consensus for discrete-time multi-agent systems and in time-inhomogeneous Markov chains. We will show how this approach could contribute to shed light to synchronizing automata theory, as for example, to generate new families of slowly synchronizing automata. Moreover, we will study the primitivity phenomenon in a probabilistic framework, presenting novel results on the randomized generation of primitive sets as
a generalized version of the classic Erdős–Rényi model.
Mercoledì 24 Maggio 2023, ore 14.30, Aula Tricerri.
Mercoledì 24 Maggio 2023, ore 14.30, Aula Tricerri.
Marco Vergani (UNIFI)
Marco Vergani (UNIFI)
Title: On Quadratic Rational Groups
Title: On Quadratic Rational Groups
Abstract: In this seminar I will talk about families of groups that have a characterization of their integral central units inside the rational group algebra. Quadratic rational groups are finite groups that have a rational field of values for any irreducible character with degree at most two over Q. From a character table perspective, they are the "dual" concept of the well-studied semirational groups.
Abstract: In this seminar I will talk about families of groups that have a characterization of their integral central units inside the rational group algebra. Quadratic rational groups are finite groups that have a rational field of values for any irreducible character with degree at most two over Q. From a character table perspective, they are the "dual" concept of the well-studied semirational groups.
Mercoledì 17 Maggio 2023, ore 14.30, Aula 230 Plesso Didattico Morgagni
Mercoledì 17 Maggio 2023, ore 14.30, Aula 230 Plesso Didattico Morgagni
Nicolas Pinzauti
Nicolas Pinzauti
Title: Critical Classes of power graphs and reconstruction of directed power graphs
Title: Critical Classes of power graphs and reconstruction of directed power graphs
Mercoledì 19 Aprile 2023, ore 14.30, Aula Tricerri.
Mercoledì 19 Aprile 2023, ore 14.30, Aula Tricerri.
Vincenzo Mantova (Leeds, UK)
Vincenzo Mantova (Leeds, UK)
Title: Solving polynomial-exponential equations, one step at a time
Title: Solving polynomial-exponential equations, one step at a time
Abstract: In the early '00s, Zilber proposed a conjectural classification of which systems of polynomial-exponential equations admit complex solutions. If true, it would amount to a "fundamental theorem of algebra" for exponentiation and have striking model theoretic consequences. I will briefly discuss the history of the problem and its relationship with Schanuel's conjecture, then focus on joint work with David Masser: we solve systems of n variables, under the assumption that the variables move on an algebraic curve, and in particular give a complete picture for the case of two variables. The proof relies on very classical results about the de Rham cohomology of curves and a modicum of differential transcendence for the exponential function.
Abstract: In the early '00s, Zilber proposed a conjectural classification of which systems of polynomial-exponential equations admit complex solutions. If true, it would amount to a "fundamental theorem of algebra" for exponentiation and have striking model theoretic consequences. I will briefly discuss the history of the problem and its relationship with Schanuel's conjecture, then focus on joint work with David Masser: we solve systems of n variables, under the assumption that the variables move on an algebraic curve, and in particular give a complete picture for the case of two variables. The proof relies on very classical results about the de Rham cohomology of curves and a modicum of differential transcendence for the exponential function.
Giovedì 30 Marzo 2023, ore 15.15, Aula 102.
Giovedì 30 Marzo 2023, ore 15.15, Aula 102.
Stacey Law (Cambridge, UK)
Stacey Law (Cambridge, UK)
Title: Sylow branching coefficients for symmetric groups
Abstract: One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.
Abstract: One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.
Giovedì 16 Marzo 2023, ore 15.30, Aula Tricerri.
Giovedì 16 Marzo 2023, ore 15.30, Aula Tricerri.
Zeinab Akhlaghi (Tehran Polytechnic, Iran)
Zeinab Akhlaghi (Tehran Polytechnic, Iran)
Title. The average of character degree and solvability of groups
Abstract. The interaction between the structure of finite groups and character degrees has been a main topic of interest for a long time. In this respect, some invariants related to character degrees have been considered and their influence on the group structure has been studied . One of these invariants is the so-called average character degree, which has recently attracted considerable interest. It is also natural to consider the average degree of certain subsets of irreducible characters. Relevant instances are the sets of rational or real characters and the set of characters whose degree is not divisible by a given prime p. In this talk, I am going to speak about some new results on this topic.
Abstract. The interaction between the structure of finite groups and character degrees has been a main topic of interest for a long time. In this respect, some invariants related to character degrees have been considered and their influence on the group structure has been studied . One of these invariants is the so-called average character degree, which has recently attracted considerable interest. It is also natural to consider the average degree of certain subsets of irreducible characters. Relevant instances are the sets of rational or real characters and the set of characters whose degree is not divisible by a given prime p. In this talk, I am going to speak about some new results on this topic.
Giovedì 9 Marzo 2023, ore 12:30, Aula 102.
Giovedì 9 Marzo 2023, ore 12:30, Aula 102.
Martino Garonzi (Universidade de Brasilia, Brasil)
Martino Garonzi (Universidade de Brasilia, Brasil)
Title. Pairwise generating sets in symmetric and alternating groups
Abstract. In this talk, I will present recent results concerning pairwise generating sets in alternating and symmetric groups, joint works with A. Maróti (Rényi Institute of Mathematics, Budapest, Hungary), F. Fumagalli and P. Gheri (University of Firenze, Italy). Given a finite 2-generated group G, a subset S of G is said to be a pairwise generating set if any two distinct elements of S generate G. We study the maximal size of a pairwise generating set, in case G is an alternating or symmetric group. This is the clique number of the so-called generating graph of G, whose vertices are the elements of G and two vertices are connected by an edge if they generate G.
Giovedì 2 Febbraio 2023, ore 15, Aula Tricerri.
Giovedì 2 Febbraio 2023, ore 15, Aula Tricerri.
Anupam Singh (IISER Pune, India)
Anupam Singh (IISER Pune, India)
Title: Matrix Waring problem
Title: Matrix Waring problem
Abstract: In this talk, we will look at the analogues question to the classical Waring problem over objects with noncommutative structures, such as groups, Lie algebras, matrix algebras, etc. We review some work done for finite groups in recent times. For example, Shalev showed that for every finite (nonabelian) simple group of sufficiently high order every element can be expressed as values of word w of length 3. This was later improved to 2 by Larsen, Shalev and Tiep. Larsen conjectured that a similar result should hold for matrices over finite fields. We have, for all integers k geq 1, there exists a constant C_k depending only on k, such that for all q > C_k and for all k \geq 1 every matrix in M_n(F_q) is a sum of two k-th powers. This work is done in collaboration with Krishna Kishore.
Giovedì 15 Dicembre 2022, ore 16.30, Aula 207.
Giovedì 15 Dicembre 2022, ore 16.30, Aula 207.
Damiano Rossi (City University London)
Damiano Rossi (City University London)
Title: Alternating sums for unipotent characters
Title: Alternating sums for unipotent characters
Abstract: Dade's conjecture predicts the number of characters of a given defect belonging in a given block in terms of certain p-local structures. We consider a version of Dade's conjecture for unipotent characters of finite reductive groups. For this purpose we replace the above mentioned p-structures with geometric analogues that are compatible with generalised Harish Chandra theory. Furthermore, we prove this new conjecture for linear and symplectic groups.
Giovedì 24 Novembre 2022, ore 15.30, Aula Tricerri.
Giovedì 24 Novembre 2022, ore 15.30, Aula Tricerri.
Margherita Paolini (Università dell'Aquila)
Margherita Paolini (Università dell'Aquila)
Title: Integral forms of affine Lie Algebras.
Title: Integral forms of affine Lie Algebras.
Abstract: Sia g un'algebra di Lie e U(g) la sua algebra inviluppante universale. Lo studio della teoria delle rappresentazioni di tipo "highest weight vector" passa attraverso la presentazione di una idonea Z-sottoalgebra di U(g) detta Forma intera delle potenze divise. Durante il talk vorrei ripercorrere lo studio delle forme intere, in primis vedremmo qualche semplice caso di tipo commutativo, passando poi al caso in cui g sia di tipo finito dimensionale. Questi esempi serviranno ad esplicare quello che accade nei casi più complessi, specificatamente quando g è una algebra di tipo affine anche di tipo torto.
Giovedì 27 Ottobre 2022, ore 15.00, Aula Tricerri.
Giovedì 27 Ottobre 2022, ore 15.00, Aula Tricerri.
Nicola Sambonet (Universidade Federal da Bahia, Brasil)
Nicola Sambonet (Universidade Federal da Bahia, Brasil)
Title: Covering groups of minimal exponent.
Title: Covering groups of minimal exponent.
Abstract: In this talk we will review some aspects of Schur's theory on projective representations, and its connection with homology discovered by Hopf. Our goal is to illustrate how, by presenting a finite group by a free product of finite cyclic groups, the Hopf formula for the Schur multiplier affords also a covering group. Remarkably, the resulting cover has minimal exponent provided that the presentation preserves the generators order, and the topology underneath this condition leads us to a beautiful interaction between an algebraic and a geometric notion.
Abstract: In this talk we will review some aspects of Schur's theory on projective representations, and its connection with homology discovered by Hopf. Our goal is to illustrate how, by presenting a finite group by a free product of finite cyclic groups, the Hopf formula for the Schur multiplier affords also a covering group. Remarkably, the resulting cover has minimal exponent provided that the presentation preserves the generators order, and the topology underneath this condition leads us to a beautiful interaction between an algebraic and a geometric notion.
Giovedì 13 Ottobre 2022, ore 15.00, Aula Tricerri.
Giovedì 13 Ottobre 2022, ore 15.00, Aula Tricerri.
Title: A version of Gallagher's theorem for real characters.
Abstract: Let N be a normal subgroup of a finite group G and let \psi be a G-invariant irreducible character of N. Gallagher showed that the number of characters of G above \psi equals the number of conjugacy classes of G/N which are good for \psi.
Title: A version of Gallagher's theorem for real characters.
Abstract: Let N be a normal subgroup of a finite group G and let \psi be a G-invariant irreducible character of N. Gallagher showed that the number of characters of G above \psi equals the number of conjugacy classes of G/N which are good for \psi.
We generalize this to determine the number of real characters of G above \psi, when the G-orbit is \psi and its dual. Consider the good conjugacy classes of G/N which are real. We show how to assign a parameter 0,+1 and -1 to each. The number of real characters of G lying over \psi is the number of real classes, weighted by this parameter.
A.A. 2021/2022
A.A. 2021/2022
Martedì 28 Giugno 2022, ore 14.30, Aula Tricerri.
Martedì 28 Giugno 2022, ore 14.30, Aula Tricerri.
Leonid Kurdachenko (Dnipro National University - Ucraina)
Leonid Kurdachenko (Dnipro National University - Ucraina)
Title: On nilpotency in groups and modules.
Title: On nilpotency in groups and modules.
Giovedì 12 Maggio 2022, ore 14.30, Aula 202 presso il Centro didattico di viale Morgagni.
Giovedì 12 Maggio 2022, ore 14.30, Aula 202 presso il Centro didattico di viale Morgagni.
Title: Babai's conjecture for classical groups with generating sets containing a transvection.
Title: Babai's conjecture for classical groups with generating sets containing a transvection.
Abstract: Let G be a finite group. A generating set of G is called symmetric if it contains the inverses of its elements. The diameter of a generating set S of G is the smallest positive integer n such that every element of G is a product of at most n elements of S. The diameter of G is the maximum diameter of a symmetric generating set of G. In other words, the diameter of G is the maximum diameter of a connected Cayley graph of G. In 1992, László Babai brilliantly conjectured that the diameter of any finite simple group G is bounded above by (log|G|)^c, where c is an absolute constant. In a joint work with Zoltán Halasi and Gábor Somlai, we proved Babai's conjecture for the projective symplectic, unitary and special linear simple groups, with few exceptions, in the special case in which the generating set S contains a transvection (which can be thought of as a nontrivial element with the smallest possible support, i.e. the largest possible set of fixed points, in its action on the vector space). In this talk I will present our result and give the main ideas of its proof. Arxiv link. https://arxiv.org/abs/2203.03323
Abstract: Let G be a finite group. A generating set of G is called symmetric if it contains the inverses of its elements. The diameter of a generating set S of G is the smallest positive integer n such that every element of G is a product of at most n elements of S. The diameter of G is the maximum diameter of a symmetric generating set of G. In other words, the diameter of G is the maximum diameter of a connected Cayley graph of G. In 1992, László Babai brilliantly conjectured that the diameter of any finite simple group G is bounded above by (log|G|)^c, where c is an absolute constant. In a joint work with Zoltán Halasi and Gábor Somlai, we proved Babai's conjecture for the projective symplectic, unitary and special linear simple groups, with few exceptions, in the special case in which the generating set S contains a transvection (which can be thought of as a nontrivial element with the smallest possible support, i.e. the largest possible set of fixed points, in its action on the vector space). In this talk I will present our result and give the main ideas of its proof. Arxiv link. https://arxiv.org/abs/2203.03323
Giovedì 5 Maggio 2022, ore 14.30, Aula Tricerri.
Giovedì 5 Maggio 2022, ore 14.30, Aula Tricerri.
Title: Powerfully nilpotent, solvable and simple groups.
Title: Powerfully nilpotent, solvable and simple groups.
Abstract: In this talk we discuss a special subclass of powerful groups called powerfully nilpotent groups. These are finite p-groups that possess a central series of a special kind. We will describe some structure theory and a ‘classification’ in terms of an ancestry tree and powerful coclass. One can view powerfully nilpotent groups as the powerful analogue of nilpotent groups. There is likewise a natural powerful analogue of solvable groups, "powerfully solvable groups", that we will also discuss briefly. For a special situation one can also introduce "powerfully simple groups".
Abstract: In this talk we discuss a special subclass of powerful groups called powerfully nilpotent groups. These are finite p-groups that possess a central series of a special kind. We will describe some structure theory and a ‘classification’ in terms of an ancestry tree and powerful coclass. One can view powerfully nilpotent groups as the powerful analogue of nilpotent groups. There is likewise a natural powerful analogue of solvable groups, "powerfully solvable groups", that we will also discuss briefly. For a special situation one can also introduce "powerfully simple groups".
Mercoledì 13 Aprile 2022, ore 15.30, Aula Tricerri.
Mercoledì 13 Aprile 2022, ore 15.30, Aula Tricerri.
Title: The non-F graph of a finite group
Title: The non-F graph of a finite group
Abstract: Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.
Given a class of groups F and a group G, we consider the graph whose vertices are the elements of G, and there is an edge between two vertices g,h in G, if <g,h> is not in F. The subgraph induced by the non-isolated vertices is the non-F graph of G. This object is a generalization of some known graphs (e.g. the non-commuting graph defined by Paul Erdos) previously studied with ad-hoc techniques which we try to put in a general framework. We investigate mainly the set of isolated vertices, which sometimes is a subgroup with an algebraic meaning and some connectivity properties. We apply these results to various notable classes and we also present some related graphs.
Abstract: Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.
Given a class of groups F and a group G, we consider the graph whose vertices are the elements of G, and there is an edge between two vertices g,h in G, if <g,h> is not in F. The subgraph induced by the non-isolated vertices is the non-F graph of G. This object is a generalization of some known graphs (e.g. the non-commuting graph defined by Paul Erdos) previously studied with ad-hoc techniques which we try to put in a general framework. We investigate mainly the set of isolated vertices, which sometimes is a subgroup with an algebraic meaning and some connectivity properties. We apply these results to various notable classes and we also present some related graphs.
Mercoledì 6 Aprile 2022, ore 15.30, Aula Tricerri.
Mercoledì 6 Aprile 2022, ore 15.30, Aula Tricerri.
Title: Characters and generation of Sylow subgroups
Title: Characters and generation of Sylow subgroups
Abstract: In this talk I will present a characterization of the groups possessing 2-generated Sylow 2-subgroups in terms of their character theory (the content is based on joint works with G. Navarro, N. Rizo and A. A. Schaeffer Fry).
Abstract: In this talk I will present a characterization of the groups possessing 2-generated Sylow 2-subgroups in terms of their character theory (the content is based on joint works with G. Navarro, N. Rizo and A. A. Schaeffer Fry).
Giovedì 10 Marzo 2022, ore 14.30, Aula Tricerri.
Giovedì 10 Marzo 2022, ore 14.30, Aula Tricerri.
Angel Garcia Blazquez (Murcia)
Angel Garcia Blazquez (Murcia)
Title: The isomorphism problem for rational groups algebras of metacyclic groups
Title: The isomorphism problem for rational groups algebras of metacyclic groups
Mercoledì 16 Febbraio 2022, ore 14.30, Aula Tricerri.
Mercoledì 16 Febbraio 2022, ore 14.30, Aula Tricerri.
Title 1: Algebre assiali
Abstract: Verranno introdotte le algebre assiali e mostrata la loro connessione con la classificazione dei gruppi semplici finiti. Faremo una panoramica dello stato attuale della ricerca, con particolare attenzione al caso delle algebre 2-generate.
Title 1: Algebre assiali
Abstract: Verranno introdotte le algebre assiali e mostrata la loro connessione con la classificazione dei gruppi semplici finiti. Faremo una panoramica dello stato attuale della ricerca, con particolare attenzione al caso delle algebre 2-generate.
Title 2: Rappresentazioni di Maiorana
Title 2: Rappresentazioni di Maiorana
Abstract: Le rappresentazioni di Majorana costituiscono un sistema assiomatico per studiare l'azione del Mostro sull’algebra di Griess. Il tema è strettamente collegato alle algebre assiali essendo l’algebra di Griess un’algebra assiale di tipo Mostro con parametri (1/4, 1/32). Anche in questo caso faremo una panoramica dello stato attuale della ricerca, dei metodi utilizzati (rappresentazioni dei gruppi simmetrici, association schemes), con particolare riguardo ai gruppi simmetrici ed alterni.
Mercoledì 26 Gennaio 2022, ore 14.30, Aula Tricerri.
Mercoledì 26 Gennaio 2022, ore 14.30, Aula Tricerri.
Yu Zeng
Yu Zeng
Title:On the proportion of vanishing elements in finite groups
Title:On the proportion of vanishing elements in finite groups
Abstract: In this talk, I will present recent results regarding the function which measuring the proportion of the elements of a finite group G that are zeros of irreducible characters of G. This is a joint work with Silvio Dolfi and Dongfang Yang.
Abstract: In this talk, I will present recent results regarding the function which measuring the proportion of the elements of a finite group G that are zeros of irreducible characters of G. This is a joint work with Silvio Dolfi and Dongfang Yang.
Mercoledì 15 Dicembre 2021, ore 14.30, Aula Tricerri.
Mercoledì 15 Dicembre 2021, ore 14.30, Aula Tricerri.
Title: Subnormality in linear groups
Title: Subnormality in linear groups
Abstract: The aim of this talk is to give an overview of the behaviour of subnormal subgroups in linear groups, and in particular of a recent work I have made in collaboration with F. de Giovanni e B.A.F. Wehrfritz.
Abstract: The aim of this talk is to give an overview of the behaviour of subnormal subgroups in linear groups, and in particular of a recent work I have made in collaboration with F. de Giovanni e B.A.F. Wehrfritz.
Giovedì 25 Novembre 2021, ore 14.30, Aula Tricerri.
Giovedì 25 Novembre 2021, ore 14.30, Aula Tricerri.
Title: Automorphism groups of regular trees: from the Euler-Poincarè characteristic to the double coset zeta-functions
Title: Automorphism groups of regular trees: from the Euler-Poincarè characteristic to the double coset zeta-functions
Abstract: The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an Euler-Poincaré characteristic: surprisingly it is no longer just a number but it is a rational multiple of a Haar measure. A key source of totally disconnected locally compact groups consists of the automorphism groups of locally finite graphs. Since the computation of the Euler-Poincarè characteristic of the automorphism group is quite gentle when the graph is a (coloured) regular tree, I plan to use such groups as leading examples throughout the talk. In particular, I will show that the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function and, because of it, I plan to conclude the talk by opening more questions than those I was able to address.
Abstract: The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an Euler-Poincaré characteristic: surprisingly it is no longer just a number but it is a rational multiple of a Haar measure. A key source of totally disconnected locally compact groups consists of the automorphism groups of locally finite graphs. Since the computation of the Euler-Poincarè characteristic of the automorphism group is quite gentle when the graph is a (coloured) regular tree, I plan to use such groups as leading examples throughout the talk. In particular, I will show that the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function and, because of it, I plan to conclude the talk by opening more questions than those I was able to address.
Based on a joint work with Gianmarco Chinello and Thomas Weigel.
Giovedì 11 Novembre 2021, ore 14.30, Aula Tricerri.
Giovedì 11 Novembre 2021, ore 14.30, Aula Tricerri.
Title: Maximal tori in HH^1 and the homotopy theory of bound quivers
Title: Maximal tori in HH^1 and the homotopy theory of bound quivers
Abstract: In this talk I will present recent results regarding the Lie algebra structure of the first Hochschild cohomology of a finite dimensional algebra. More precisely, I will explain how to classify the maximal tori in this Lie algebra using the homotopy theory of quivers. Time permitting, I will provide various applications to monomial algebras, blocks with cyclic defect groups and group algebras of p-groups. This is all joint work with Benjamin Briggs.
Abstract: In this talk I will present recent results regarding the Lie algebra structure of the first Hochschild cohomology of a finite dimensional algebra. More precisely, I will explain how to classify the maximal tori in this Lie algebra using the homotopy theory of quivers. Time permitting, I will provide various applications to monomial algebras, blocks with cyclic defect groups and group algebras of p-groups. This is all joint work with Benjamin Briggs.
Giovedì 4 Novembre 2021, ore 14.30, Aula Tricerri.
Giovedì 4 Novembre 2021, ore 14.30, Aula Tricerri.
Title: On character degrees
Title: On character degrees
Abstract: Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement.
Abstract: Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement.
We present a characterization, under some separability conditions, of when Irrπ(G) = Irrρ(G) where π and ρ are sets of primes and Irrπ(G) is the set of irreducible characters χ of G such that all the primes dividing χ(1) lie in π. This generalizes the well-known theorems of Ito-Michler and Thompson on character degrees. We will also present a recent version for Brauer characters in p-solvable groups.
This is joint work with L. Bonazzi, G. Navarro and N. Rizo.
Giovedì 28 Ottobre 2021, ore 14.30, Aula Tricerri.
Giovedì 28 Ottobre 2021, ore 14.30, Aula Tricerri.
Title: Fusion systems on maximal class p-groups
Title: Fusion systems on maximal class p-groups
Abstract: Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. Given a finite group G, it is always possible to define the saturated fusion system realized by G on one of its Sylow p-subgroups. However, not all saturated fusion systems can be realized in this way. When this is the case, we say that the fusion system is exotic. The understanding of the behavior of exotic fusion systems (in particular at odd primes) is still an important open problem. It turns out that many of the known exotic fusion systems are defined on p-groups of maximal nilpotency class. In this talk we will present the classification of saturated fusion systems on p-groups of maximal nilpotency class, highlighting its relevance for the study of exotic fusion systems.
Abstract: Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. Given a finite group G, it is always possible to define the saturated fusion system realized by G on one of its Sylow p-subgroups. However, not all saturated fusion systems can be realized in this way. When this is the case, we say that the fusion system is exotic. The understanding of the behavior of exotic fusion systems (in particular at odd primes) is still an important open problem. It turns out that many of the known exotic fusion systems are defined on p-groups of maximal nilpotency class. In this talk we will present the classification of saturated fusion systems on p-groups of maximal nilpotency class, highlighting its relevance for the study of exotic fusion systems.
A.A. 2020/2021
A.A. 2020/2021
Mercoledì 30 Giugno 2021, ore 14.30, Online.
Mercoledì 30 Giugno 2021, ore 14.30, Online.
Dario Villanis Ziani (Firenze)
Dario Villanis Ziani (Firenze)
Title: Just infinite profinite structures
Title: Just infinite profinite structures
Abstract:
In this talk we will examine some topics of my PhD thesis.
Abstract:
In this talk we will examine some topics of my PhD thesis.
In the first part, we will look at just infinite profinite groups. For long time, only just infinite pro-p groups were known. Thus, we will exhibit some new examples of profinite just infinite groups that are not pro-p groups.
In the second part of the talk, we will examine other just infinite profinite structures, in particular just infinite profinite Lie rings, in order to find some characterization theorems similar to the results established by C. Reid for just infinite profinite groups; more explicitely, we will provide a full characterization for Lie rings in a suitable abstract class, giving also some concrete examples.
Mercoledì 23 Giugno 2021, ore 14.30, Online.
Mercoledì 23 Giugno 2021, ore 14.30, Online.
Giada Volpato (Firenze)
Giada Volpato (Firenze)
Title: Non-linear Sylow branching coefficients for S_n.
Title: Non-linear Sylow branching coefficients for S_n.
Mercoledì 16 Giugno 2021, ore 14.30, Online.
Mercoledì 16 Giugno 2021, ore 14.30, Online.
Luca Sabatini (Firenze)
Luca Sabatini (Firenze)
Title: The growth of abelian sections.
Title: The growth of abelian sections.
Mercoledì 9 Giugno 2021, ore 14.30, Online.
Mercoledì 9 Giugno 2021, ore 14.30, Online.
Lorenzo Bonazzi (Firenze)
Lorenzo Bonazzi (Firenze)
Title: Finite groups in which the real character degrees and the real class sizes are prime powers.
Title: Finite groups in which the real character degrees and the real class sizes are prime powers.
Mercoledì 26 Maggio 2021, ore 14.30, Online.
Mercoledì 26 Maggio 2021, ore 14.30, Online.
Title: Coprime partitions and Jordan totient functions.
Title: Coprime partitions and Jordan totient functions.
Martedì 11 Maggio 2021, ore 16.30, Online.
Martedì 11 Maggio 2021, ore 16.30, Online.
Title: An upper bound for the nonsolvable length of a finite group in terms of its shortest law.
Title: An upper bound for the nonsolvable length of a finite group in terms of its shortest law.
A.A. 2019/2020
A.A. 2019/2020
Mercoledì 10 Giugno 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 10 Giugno 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 27 Maggio 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 27 Maggio 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 13 Maggio 2020, ore 14.30, Aula (ANNULLATO)
Mercoledì 13 Maggio 2020, ore 14.30, Aula (ANNULLATO)
Mercoledì 22 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 22 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 15 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 15 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Lleonard Rubio (Padova)
Lleonard Rubio (Padova)
Mercoledì 8 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Mercoledì 8 Aprile 2020, ore 14.30, Aula Tricerri (ANNULLATO)
Victor Ortiz Sotomayor (Valencia)
Victor Ortiz Sotomayor (Valencia)
Mercoledì 25 Marzo 2020, ore 14.30, Aula 2 (ANNULLATO)
Mercoledì 25 Marzo 2020, ore 14.30, Aula 2 (ANNULLATO)
Mercoledì 18 Marzo 2020, ore 14.30, Aula 2 (ANNULLATO)
Mercoledì 18 Marzo 2020, ore 14.30, Aula 2 (ANNULLATO)
Title: Generalised Right-Angled Artin pro-p groups lab–rats for arithmetic conjectures
Title: Generalised Right-Angled Artin pro-p groups lab–rats for arithmetic conjectures
Mercoledì 4 Marzo 2020, ore 14.30, Aula Tricerri
Mercoledì 4 Marzo 2020, ore 14.30, Aula Tricerri
Claudio Marchi (Manchester)
Claudio Marchi (Manchester)
Title: Picard groups and Picent of blocks.
Title: Picard groups and Picent of blocks.
Abstract: The Picard group of a block over a local ring is a particularly interesting object, as shown recently by Boltje, Kessar and Linckelmann, and has been used to great effect to deal with Donovan's conjecture. For this, and other reasons, much work is being done to study the structure of Picard groups, as in recent papers by Eisele and Livesey. In this talk I will give an introduction to Picard groups and I will present joint work with Michael Livesey on Picent of blocks, the subgroup of the Picard group given by Morita auto-equivalences inducing the trivial permutation on irreducible characters lying in the block.
Abstract: The Picard group of a block over a local ring is a particularly interesting object, as shown recently by Boltje, Kessar and Linckelmann, and has been used to great effect to deal with Donovan's conjecture. For this, and other reasons, much work is being done to study the structure of Picard groups, as in recent papers by Eisele and Livesey. In this talk I will give an introduction to Picard groups and I will present joint work with Michael Livesey on Picent of blocks, the subgroup of the Picard group given by Morita auto-equivalences inducing the trivial permutation on irreducible characters lying in the block.
Mercoledì 26 Febbraio 2020, ore 14.30, Aula 2
Mercoledì 26 Febbraio 2020, ore 14.30, Aula 2
Title: On groups with restrictions on subnormal subgroups
Title: On groups with restrictions on subnormal subgroups
Abstract: In the universe of infinite groups, I will discuss properties which generalizes the normality of a subgroup. In particular, I will consider cn-subgroups, i.e. subgroups which are commensurable with some normal subgroup and describe generalized soluble groups in which "many" subgroups are cn, where "many" may mean:
Abstract: In the universe of infinite groups, I will discuss properties which generalizes the normality of a subgroup. In particular, I will consider cn-subgroups, i.e. subgroups which are commensurable with some normal subgroup and describe generalized soluble groups in which "many" subgroups are cn, where "many" may mean:
- all subgroups are cn,
- all subgroups are cn,
- all subnormal subgroups are cn,
- all subnormal subgroups are cn,
- the set of non-cn subgroups satisfies the weak minimal condition.
- the set of non-cn subgroups satisfies the weak minimal condition.
Mercoledì 19 Febbraio 2020, ore 14.30, Aula Tricerri
Mercoledì 19 Febbraio 2020, ore 14.30, Aula Tricerri
Carlo Toffalori (Camerino)
Carlo Toffalori (Camerino)
Title: Some model theory of modules over algebraic integers and pairs of Dedekind domains
Title: Some model theory of modules over algebraic integers and pairs of Dedekind domains
Abstract: I will introduce a joint work with Lorna Gregory, Ivo Herzog and Sonia L'Innocente. Our long-term program regards the model theory of modules over the ring A of algebraic integers. It is known that their first order theory is decidable but the proof (by Gena Puninski, Sonia L'Innocente and myself) does note provide crucial model theoretic information. On the other hand, A can be obtained as a direct union of Dedekind domains, that is, number fields. Hence we focus on extensions of these domains and we examine several model theoretic problems about them.
Abstract: I will introduce a joint work with Lorna Gregory, Ivo Herzog and Sonia L'Innocente. Our long-term program regards the model theory of modules over the ring A of algebraic integers. It is known that their first order theory is decidable but the proof (by Gena Puninski, Sonia L'Innocente and myself) does note provide crucial model theoretic information. On the other hand, A can be obtained as a direct union of Dedekind domains, that is, number fields. Hence we focus on extensions of these domains and we examine several model theoretic problems about them.
Mercoledì 12 Febbraio 2020, ore 14.30, Aula Tricerri
Mercoledì 12 Febbraio 2020, ore 14.30, Aula Tricerri
Title: Synchronization
Title: Synchronization
Mercoledì 5 Febbraio 2020, ore 14.30, Aula Tricerri
Mercoledì 5 Febbraio 2020, ore 14.30, Aula Tricerri
Title: Conway’s surreal numbers, asymptotic analysis and transseries
Title: Conway’s surreal numbers, asymptotic analysis and transseries
Abstract: We give a presentation of Conway’s surreal numbers which emphasizes the connections with asymptotic analysis and transseries. Both the surreals and transseries extend the field of Laurent series, but, unlike the Laurent series, they are also closed under exponentiation and formal integration. Applications to a problem of Skolem on exponential polynomials will also be discussed.
Abstract: We give a presentation of Conway’s surreal numbers which emphasizes the connections with asymptotic analysis and transseries. Both the surreals and transseries extend the field of Laurent series, but, unlike the Laurent series, they are also closed under exponentiation and formal integration. Applications to a problem of Skolem on exponential polynomials will also be discussed.
Mercoledì 29 Gennaio 2020, ore 14.30, Aula 2
Mercoledì 29 Gennaio 2020, ore 14.30, Aula 2
Title: Trapezoidal words
Title: Trapezoidal words
Mercoledì 22 Gennaio 2020, ore 14.30, Aula Tricerri
Mercoledì 22 Gennaio 2020, ore 14.30, Aula Tricerri
Title: On Thompson's normal p-complement theorem
Title: On Thompson's normal p-complement theorem
Mercoledì 15 Gennaio 2020, Università di Bologna – Dipartimento di Matematica.
Mercoledì 15 Gennaio 2020, Università di Bologna – Dipartimento di Matematica.
(Piazza di Porta S. Donato, 5, 40126 Bologna BO)
(Piazza di Porta S. Donato, 5, 40126 Bologna BO)
Title: The road closure conjecture
Title: The road closure conjecture
Abstract: There is a new approach that has been proposed recently for the study of certain regular semigroups. This new approach involves the study of finite primitive groups. Via this method, many problems in the context of regular semigroups translate into natural problems in finite primitive groups. We give some details concerning the relation between regular semigroups and finite primitive groups. In particular, we discuss the Road Closure Conjecture on finite primitive groups, which is strictly related to a possible classification of certain idempotent generated regular semigroups.
Abstract: There is a new approach that has been proposed recently for the study of certain regular semigroups. This new approach involves the study of finite primitive groups. Via this method, many problems in the context of regular semigroups translate into natural problems in finite primitive groups. We give some details concerning the relation between regular semigroups and finite primitive groups. In particular, we discuss the Road Closure Conjecture on finite primitive groups, which is strictly related to a possible classification of certain idempotent generated regular semigroups.
Mercoledì 18 Dicembre 2019, ore 14.30, Aula 2.
Mercoledì 18 Dicembre 2019, ore 14.30, Aula 2.
Damiano Rossi (Wuppertal)
Damiano Rossi (Wuppertal)
Title: Character triple conjecture for p-solvable groups
Title: Character triple conjecture for p-solvable groups
Mercoledì 11 Dicembre 2019, Lauree, Niente Seminario.
Mercoledì 11 Dicembre 2019, Lauree, Niente Seminario.
Mercoledì 4 Dicembre 2019, ore 14.30, Aula Tricerri.
Mercoledì 4 Dicembre 2019, ore 14.30, Aula Tricerri.
Title: Fixed points, abelianizations and amalgamations: linear groups and unit groups of group rings
Title: Fixed points, abelianizations and amalgamations: linear groups and unit groups of group rings
Mercoledì 27 Novembre 2019, ore 14.30, Aula 2.
Mercoledì 27 Novembre 2019, ore 14.30, Aula 2.
Title: Metaquasihamiltonian groups and related topics
Title: Metaquasihamiltonian groups and related topics
Mercoledì 20 Novembre 2019, ore 14.30, Aula 2.
Mercoledì 20 Novembre 2019, ore 14.30, Aula 2.
Title: Orders of units in group rings and blocks of defect 1
Title: Orders of units in group rings and blocks of defect 1
Mercoledì 13 Novembre 2019, ore 14.30, Aula 2.
Mercoledì 13 Novembre 2019, ore 14.30, Aula 2.
Title: Galois action on the principal block
Title: Galois action on the principal block
Mercoledì 6 Novembre 2019, ore 14.30, Aula 2.
Mercoledì 6 Novembre 2019, ore 14.30, Aula 2.
Title: Invariable generation of groups
Title: Invariable generation of groups
Mercoledì 30 Ottobre 2019, niente Seminario.
Mercoledì 30 Ottobre 2019, niente Seminario.
Mercoledì 23 Ottobre 2019, ore 14.30, Aula 2.
Mercoledì 23 Ottobre 2019, ore 14.30, Aula 2.
Sabino Di Trani (Firenze)
Sabino Di Trani (Firenze)
Title: A Reeder’s Conjecture on small representations in the Exterior Algebra
Title: A Reeder’s Conjecture on small representations in the Exterior Algebra
Abstract: A well known result of the first half of XX century asserts that the cohomology of a compact connected Lie group over C is isomorphic as graded vector space to the ring of G-invariants in the exterior algebra of g=Lie(G). Finding Betti numbers of G corresponds then to locate copies of trivial representations in \Lambda g. Reeder educes this computation to a problem of finite group representations, involving the Weyl group of G. Generalizing this result, he conjectures that analogously it is possible to locate some special irreducible representations of G, called “small representations”, reducing to a similar “Weyl group Representation” – problem. In the talk I will analyze the framework of Reeder’s Conjecture and present some strategies about how to prove the conjecture in case of Classical Lie groups of type B and C.
Abstract: A well known result of the first half of XX century asserts that the cohomology of a compact connected Lie group over C is isomorphic as graded vector space to the ring of G-invariants in the exterior algebra of g=Lie(G). Finding Betti numbers of G corresponds then to locate copies of trivial representations in \Lambda g. Reeder educes this computation to a problem of finite group representations, involving the Weyl group of G. Generalizing this result, he conjectures that analogously it is possible to locate some special irreducible representations of G, called “small representations”, reducing to a similar “Weyl group Representation” – problem. In the talk I will analyze the framework of Reeder’s Conjecture and present some strategies about how to prove the conjecture in case of Classical Lie groups of type B and C.
Mercoledì 16 Ottobre 2019, ore 14.30, Aula 2.
Mercoledì 16 Ottobre 2019, ore 14.30, Aula 2.
Title: Classifying 2-blocks with an elementary abelian defect group
Title: Classifying 2-blocks with an elementary abelian defect group
Mercoledì 9 Ottobre 2019, ore 14.30, Aula 2.
Mercoledì 9 Ottobre 2019, ore 14.30, Aula 2.
Title: A polynomial bound of the number of maximal systems of imprimitivity of a finite transitive permutation group
Title: A polynomial bound of the number of maximal systems of imprimitivity of a finite transitive permutation group
Mercoledì 2 Ottobre 2019, ore 14.30, Aula 2.
Mercoledì 2 Ottobre 2019, ore 14.30, Aula 2.
Nicola Grittini (Firenze)
Nicola Grittini (Firenze)
Title: Characters of p-solvable and π-separable groups
Title: Characters of p-solvable and π-separable groups
A.A. 2018/2019
A.A. 2018/2019
Martedì 18 Giugno 2019. Cena di Fine di Algebra
Martedì 18 Giugno 2019. Cena di Fine di Algebra
Mercoledì 5 Giugno 2019, ore 14.30, Aula 2.
Mercoledì 5 Giugno 2019, ore 14.30, Aula 2.
Title: Integrals of Groups
Title: Integrals of Groups
Abstract: Given a group $G$, is it possible to find a group $H$ such that the commutator subgroup of $H$ is equal to $G$? This is an instance of the following general problem: given a group “functor” $F$; what can be said about the inverse image under $F$ of a given group? In many cases the corresponding inverse theory is trivial, but in some cases more can be said. We will start going over integrals of groups and discussing cases where they do exist (or cannot exist), if these integrals have to be finite and how big or small they can be. We will also discuss other related inverse group theory questions and current directions.
Abstract: Given a group $G$, is it possible to find a group $H$ such that the commutator subgroup of $H$ is equal to $G$? This is an instance of the following general problem: given a group “functor” $F$; what can be said about the inverse image under $F$ of a given group? In many cases the corresponding inverse theory is trivial, but in some cases more can be said. We will start going over integrals of groups and discussing cases where they do exist (or cannot exist), if these integrals have to be finite and how big or small they can be. We will also discuss other related inverse group theory questions and current directions.
This talk is based on joint work with João Araújo (Universidade Aberta), Peter Cameron (University of St. Andrews) and Carlo Casolo (Università di Firenze).
This talk is based on joint work with João Araújo (Universidade Aberta), Peter Cameron (University of St. Andrews) and Carlo Casolo (Università di Firenze).
Mercoledì 29 Maggio 2019, ore 14.30, Aula ?.
Mercoledì 29 Maggio 2019, ore 14.30, Aula ?.
Title: A brief survey of non-abelian tensor products of groups
Title: A brief survey of non-abelian tensor products of groups
Abstract: We survey work on non-abelian tensor products of groups, with an emphasis on non-abelian tensor squares, including both general structure results and methods for computing such groups.
Abstract: We survey work on non-abelian tensor products of groups, with an emphasis on non-abelian tensor squares, including both general structure results and methods for computing such groups.
Mercoledì 22 Maggio 2019, ore 14.30, Aula Tricerri.
Mercoledì 22 Maggio 2019, ore 14.30, Aula Tricerri.
Title: Markov chains and random walks on graphs.
Title: Markov chains and random walks on graphs.
Abstract: We start with an introduction on Markov Chains: joint distribution, states classification, invariant probabilities and Markov Theorem that establishes the asymptotic properties of the Markov chains. We define Simple Random Walks on graphs giving many examples and random walks with drift. Moreover, we will deduce asymptotic properties of those random walks.
Abstract: We start with an introduction on Markov Chains: joint distribution, states classification, invariant probabilities and Markov Theorem that establishes the asymptotic properties of the Markov chains. We define Simple Random Walks on graphs giving many examples and random walks with drift. Moreover, we will deduce asymptotic properties of those random walks.
Mercoledì 15 Maggio 2019, ore 14.30, Aula 2.
Mercoledì 15 Maggio 2019, ore 14.30, Aula 2.
Title: Uniform domination for simple groups
Title: Uniform domination for simple groups
Abstract: It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class C such that for each nontrivial element x there exists an element y in C such that x and y generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset S of conjugate elements such that for each nontrivial element x there exists an element s in S such that x and s generate the group. This invariant arises naturally in the study of generating graphs. In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods together with recent results on the base sizes of primitive permutation groups.
Abstract: It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class C such that for each nontrivial element x there exists an element y in C such that x and y generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset S of conjugate elements such that for each nontrivial element x there exists an element s in S such that x and s generate the group. This invariant arises naturally in the study of generating graphs. In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods together with recent results on the base sizes of primitive permutation groups.
Mercoledì 8 Maggio 2019, ore 14.30, Aula Tricerri.
Mercoledì 8 Maggio 2019, ore 14.30, Aula Tricerri.
Title: Elementi totalmente commutativi nei gruppi di Coxeter
Title: Elementi totalmente commutativi nei gruppi di Coxeter
Abstract: Sia W un gruppo di Coxeter. Un elemento w di W è totalmente commutativo (TC) se prese due qualsiasi delle sue espressioni ridotte si può passare dall’una all’altra soltanto con una serie di scambi di generatori che commutano. Gli elementi TC sono stati studiati nel caso finito da Stembridge e indicizzano una base dell’algebra generalizzata di Temperley-Lieb associata a W. In questo talk daremo una classificazione degli elementi TC nel caso dei gruppi di Coxeter affini e mostreremo nel caso finito di tipo A e in quello affine di tipo \tilde{A} delle interpretazioni combinatorie che permettono il calcolo della funzione generatrice degli elementi TC secondo la lunghezza di Coxeter.
Abstract: Sia W un gruppo di Coxeter. Un elemento w di W è totalmente commutativo (TC) se prese due qualsiasi delle sue espressioni ridotte si può passare dall’una all’altra soltanto con una serie di scambi di generatori che commutano. Gli elementi TC sono stati studiati nel caso finito da Stembridge e indicizzano una base dell’algebra generalizzata di Temperley-Lieb associata a W. In questo talk daremo una classificazione degli elementi TC nel caso dei gruppi di Coxeter affini e mostreremo nel caso finito di tipo A e in quello affine di tipo \tilde{A} delle interpretazioni combinatorie che permettono il calcolo della funzione generatrice degli elementi TC secondo la lunghezza di Coxeter.
Mercoledì 24 Aprile 2019, ore 14.30, Aula Tricerri.
Mercoledì 24 Aprile 2019, ore 14.30, Aula Tricerri.
Victor Ortiz-Sotomayor (Valencia)
Victor Ortiz-Sotomayor (Valencia)
Title: Conjugacy classes and π-structure of factorised groups
Title: Conjugacy classes and π-structure of factorised groups
Martedì 16 Aprile 2019, ore 14.30, Aula Tricerri.
Martedì 16 Aprile 2019, ore 14.30, Aula Tricerri.
Title: Moment graph and the combinatorics of Kazhdan-Lusztig polynomials
Title: Moment graph and the combinatorics of Kazhdan-Lusztig polynomials
Abstract: Kazhdan-Lusztig polynomials were introduced in the ’70s and, although their definition is an elementary recursive formula, they provide a deep relations between geometry and representation theory: for example they allow us to compute characters in the category O of representations of complex Lie algebras. These polynomials also possess remarkable combinatorial properties, and a open long-standing conjecture states that they only depend on their underlying poset. We will discuss how, using the point of view of moment graphs, we can shed some light on the combinatorics of these polynomials, leading us to understand the coefficient of q via an explicit formula.
Abstract: Kazhdan-Lusztig polynomials were introduced in the ’70s and, although their definition is an elementary recursive formula, they provide a deep relations between geometry and representation theory: for example they allow us to compute characters in the category O of representations of complex Lie algebras. These polynomials also possess remarkable combinatorial properties, and a open long-standing conjecture states that they only depend on their underlying poset. We will discuss how, using the point of view of moment graphs, we can shed some light on the combinatorics of these polynomials, leading us to understand the coefficient of q via an explicit formula.
Mercoledì 10 Aprile 2019, ore 14.30, Aula 2.
Mercoledì 10 Aprile 2019, ore 14.30, Aula 2.
Bianca Lodà (South Wales)
Bianca Lodà (South Wales)
Title: On the relational complexity for finite primitive almost simple groups.
Title: On the relational complexity for finite primitive almost simple groups.
Mercoledì 3 Aprile 2019, ore 14.30, Aula 2.
Mercoledì 3 Aprile 2019, ore 14.30, Aula 2.
Title: The prime graph on class sizes of a finite group has a bipartite complement
Title: The prime graph on class sizes of a finite group has a bipartite complement
Abstract: Let G be a finite group, and let cs(G) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs(G), that we denote by ∆(G), is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cs(G). In the talk we will show that ∆(G) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V(G) of ∆(G) is covered by two subsets, each inducing a complete subgraph.
Abstract: Let G be a finite group, and let cs(G) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs(G), that we denote by ∆(G), is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cs(G). In the talk we will show that ∆(G) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V(G) of ∆(G) is covered by two subsets, each inducing a complete subgraph.
Mercoledì 27 Marzo 2019, ore 14.30, Aula 2.
Mercoledì 27 Marzo 2019, ore 14.30, Aula 2.
Title: On the Structure of Finite Groups of Characterisitc p
Title: On the Structure of Finite Groups of Characterisitc p
Mercoledì 20 Marzo 2019, ore 14.30, Aula 2.
Mercoledì 20 Marzo 2019, ore 14.30, Aula 2.
Title: The Bruhat order on Abelian nilradicals and generalizations
Title: The Bruhat order on Abelian nilradicals and generalizations
Abstract: Given a simple algebraic group G and a parabolic subgroup in G with abelian unipotent radical P^u, I will consider the action of a Borel subgroup B contained in P on P^u and on its Lie algebra. For example, if B(n) denotes the subgroup of upper triangular matrices in GL(n), this includes the action of B(n) x B(m) on the space of matrices Mat(m,n) by left and right multiplication. I will explain how P^u (or equivalently its Lie algebra) decomposes into B-orbits, and decribe the partial order among the B-orbits defined by the inclusions of their closures. Then I will discuss generalizations to the context of symmetric varieties and to that of abelian ideals of Borel subalgebras. The talk is based on joint works with A. Maffei, P. Moseneder Frajria and P. Papi.
Abstract: Given a simple algebraic group G and a parabolic subgroup in G with abelian unipotent radical P^u, I will consider the action of a Borel subgroup B contained in P on P^u and on its Lie algebra. For example, if B(n) denotes the subgroup of upper triangular matrices in GL(n), this includes the action of B(n) x B(m) on the space of matrices Mat(m,n) by left and right multiplication. I will explain how P^u (or equivalently its Lie algebra) decomposes into B-orbits, and decribe the partial order among the B-orbits defined by the inclusions of their closures. Then I will discuss generalizations to the context of symmetric varieties and to that of abelian ideals of Borel subalgebras. The talk is based on joint works with A. Maffei, P. Moseneder Frajria and P. Papi.
Mercoledì 13 Marzo 2019, ore 14.30, Aula 5.
Mercoledì 13 Marzo 2019, ore 14.30, Aula 5.
Title: Crowns in finite groups: Theory and applications
Title: Crowns in finite groups: Theory and applications
Abstract: The notion of a crown in a finite group was introduced by Gaschutz in the 1960s, and later developed by Dalla Volta and Lucchini as a tool to study minimal generation and the presentation rank in finite and profinite groups. In this talk, we will give an introduction to the theory, and detail some recent applications to generation and enumeration problems in certain classes of finite groups. One application is ongoing joint work with Colva Roney-Dougal, while another is joint work with Andrea Lucchini and Claude Marion.
Abstract: The notion of a crown in a finite group was introduced by Gaschutz in the 1960s, and later developed by Dalla Volta and Lucchini as a tool to study minimal generation and the presentation rank in finite and profinite groups. In this talk, we will give an introduction to the theory, and detail some recent applications to generation and enumeration problems in certain classes of finite groups. One application is ongoing joint work with Colva Roney-Dougal, while another is joint work with Andrea Lucchini and Claude Marion.
Mercoledì 6 Marzo 2019, Lauree, Niente seminario.
Mercoledì 6 Marzo 2019, Lauree, Niente seminario.
Mercoledì 27 Febbraio 2019, ore 14.30, Aula 2.
Mercoledì 27 Febbraio 2019, ore 14.30, Aula 2.
Title: Enumerating characters of Sylow p-subgroups of finite groups of Lie type G(p^f)
Title: Enumerating characters of Sylow p-subgroups of finite groups of Lie type G(p^f)
Abstract: Let q=p^f with p a prime. The problem of enumerating characters of subgroups of a finite group of Lie type G(q) plays an important role in various research problems, from random walks on G(q) to cross-characteristics representations of G(q). O’Brien and Voll have recently determined a formula for the generic number of irreducible characters of a fixed degree of a Sylow p-subgroup U(q) of G(q), provided p>c where c is the nilpotency class of G(q). We discuss in this talk the situation in the case p smaller than or equal to c. In particular, we describe an algorithm for the parametrization of the irreducible characters of U(q) which replaces the Kirillov orbit used in the case p>c. Moreover, we present connections with a conjecture of Higman and we highlight a departure from the case of large p. This is based on joint works with Goodwin, Le and Magaard.
Abstract: Let q=p^f with p a prime. The problem of enumerating characters of subgroups of a finite group of Lie type G(q) plays an important role in various research problems, from random walks on G(q) to cross-characteristics representations of G(q). O’Brien and Voll have recently determined a formula for the generic number of irreducible characters of a fixed degree of a Sylow p-subgroup U(q) of G(q), provided p>c where c is the nilpotency class of G(q). We discuss in this talk the situation in the case p smaller than or equal to c. In particular, we describe an algorithm for the parametrization of the irreducible characters of U(q) which replaces the Kirillov orbit used in the case p>c. Moreover, we present connections with a conjecture of Higman and we highlight a departure from the case of large p. This is based on joint works with Goodwin, Le and Magaard.
Mercoledì 20 Febbraio 2019, ore 14.30, Aula 2.
Mercoledì 20 Febbraio 2019, ore 14.30, Aula 2.
Title: The McKay conjecture and fields of values of characters
Title: The McKay conjecture and fields of values of characters
Abstract: The celebrated McKay conjecture (that now is a theorem in the case where p = 2) was generalized in 2004 by G. Navarro taking into account fields of values of characters. The so-called Galois-McKay conjecture has many deep implications, nevertheless it was the last counting global/local conjecture without a corresponding reduction theorem to simple groups. In this talk I will discuss fields of values of characters involved in the McKay conjecture. I will also present a reduction theorem for the Galois-McKay conjecture proven jointly with G. Navarro and B. Späth.
Abstract: The celebrated McKay conjecture (that now is a theorem in the case where p = 2) was generalized in 2004 by G. Navarro taking into account fields of values of characters. The so-called Galois-McKay conjecture has many deep implications, nevertheless it was the last counting global/local conjecture without a corresponding reduction theorem to simple groups. In this talk I will discuss fields of values of characters involved in the McKay conjecture. I will also present a reduction theorem for the Galois-McKay conjecture proven jointly with G. Navarro and B. Späth.
Mercoledì 13 Febbraio 2019, ore 14.30, Aula 2.
Mercoledì 13 Febbraio 2019, ore 14.30, Aula 2.
Title: Gruppi che hanno lo stesso olomorfo
Title: Gruppi che hanno lo stesso olomorfo
Mercoledì 30 Gennaio 2019, ore 14.30, Aula 2.
Mercoledì 30 Gennaio 2019, ore 14.30, Aula 2.
Title: Embedding Properties in Uncountable Groups
Title: Embedding Properties in Uncountable Groups
Abstract: The aim of this talk is to show that in uncountable groups, as well as in groups of infinite rank, the behaviour of small subgroups is often neglectable.
Abstract: The aim of this talk is to show that in uncountable groups, as well as in groups of infinite rank, the behaviour of small subgroups is often neglectable.
Mercoledì 23 Gennaio 2019, ore 14.30, Aula 2.
Mercoledì 23 Gennaio 2019, ore 14.30, Aula 2.
Title: Sheaves on the alcoves and modular representations
Title: Sheaves on the alcoves and modular representations
Abstract: I’ll report on a joint project with Peter Fiebig. The aim of the project is to provide a new perspective on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. Given a finite root system R and a field k we introduce an exact category C of sheaves on the partially ordered set of alcoves associated with R, and we show that the indecomposable projective objects in C encode the desired characters.
Abstract: I’ll report on a joint project with Peter Fiebig. The aim of the project is to provide a new perspective on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. Given a finite root system R and a field k we introduce an exact category C of sheaves on the partially ordered set of alcoves associated with R, and we show that the indecomposable projective objects in C encode the desired characters.
Mercoledì 5 Dicembre 2018, ore 14.30, Aula 205 Plesso Viale Morgagni.
Mercoledì 5 Dicembre 2018, ore 14.30, Aula 205 Plesso Viale Morgagni.
Title: Hausdorff dimension in finitely generated pro-p groups
Title: Hausdorff dimension in finitely generated pro-p groups
Abstract: Every finitely generated pro-p group G comes equipped with a range of translation-invariant metrics that are naturally induced by filtration series such as the p-power filtration or the lower p-series. Given such a metric, the distribution of closed subgroups in G gives rise to a corresponding Hausdorff spectrum. It is a long-standing open question whether the finiteness of the Hausdorff spectrum, with respect to the p-power filtration, say, implies that the pro-p group G is p-adic analytic. Several related and independent questions about Hausdorff spectra of pro-p groups were raised by Aner Shalev in his `New Horizons’ survey, published in 2000. In my talk I will give an introduction to the subject and then report on joint work (i) with Anitha Thillaisundaram and Amaia Zugadi-Reizabal, (ii) with Anitha Thillaisundaram, and (iii) with Alejandra Garrido and Oihana Garaialde-Ocana — or a subset thereof, if time is short. In each of the three projects we answer some of the questions raised almost twenty years ago, but also stumble upon interesting new problems.
Abstract: Every finitely generated pro-p group G comes equipped with a range of translation-invariant metrics that are naturally induced by filtration series such as the p-power filtration or the lower p-series. Given such a metric, the distribution of closed subgroups in G gives rise to a corresponding Hausdorff spectrum. It is a long-standing open question whether the finiteness of the Hausdorff spectrum, with respect to the p-power filtration, say, implies that the pro-p group G is p-adic analytic. Several related and independent questions about Hausdorff spectra of pro-p groups were raised by Aner Shalev in his `New Horizons’ survey, published in 2000. In my talk I will give an introduction to the subject and then report on joint work (i) with Anitha Thillaisundaram and Amaia Zugadi-Reizabal, (ii) with Anitha Thillaisundaram, and (iii) with Alejandra Garrido and Oihana Garaialde-Ocana — or a subset thereof, if time is short. In each of the three projects we answer some of the questions raised almost twenty years ago, but also stumble upon interesting new problems.
Mercoledì 28 Novembre 2018, ore 14.30, Aula 2.
Mercoledì 28 Novembre 2018, ore 14.30, Aula 2.
Titolo: Baumann-components of finite groups of characteristic $p$.
Titolo: Baumann-components of finite groups of characteristic $p$.
Abstract: In the seminar I will introduce a class $Bau_p(G)$ of $p$-subgroups of finite groups $G$ of characteristic $p$. The “minimal” obstructions for $B\in Bau_p(G)$ to be normal in $G$ are called Baumann-components of $G$. I will discuss some results about Baumann-components obtained together with Ulrich Meierfrankenfeld and Bernd Stellmacher.
Abstract: In the seminar I will introduce a class $Bau_p(G)$ of $p$-subgroups of finite groups $G$ of characteristic $p$. The “minimal” obstructions for $B\in Bau_p(G)$ to be normal in $G$ are called Baumann-components of $G$. I will discuss some results about Baumann-components obtained together with Ulrich Meierfrankenfeld and Bernd Stellmacher.
Mercoledì 21 Novembre 2018, ore 14.30, Aula 2.
Mercoledì 21 Novembre 2018, ore 14.30, Aula 2.
Titolo: the arithmetic of group representations, group actions and torsion subgroups of elliptic curves
Titolo: the arithmetic of group representations, group actions and torsion subgroups of elliptic curves
Abstract: Arithmetic aspects of representations of finite groups are discussed. We present an overview of representations of finite groups, arithmetic groups with Galois action, related finite group schemes, quadratic lattices, finiteness theorems, Galois representations, Galois cohomology, representations arising from torsion points of elliptic curves and local-global principles.
Abstract: Arithmetic aspects of representations of finite groups are discussed. We present an overview of representations of finite groups, arithmetic groups with Galois action, related finite group schemes, quadratic lattices, finiteness theorems, Galois representations, Galois cohomology, representations arising from torsion points of elliptic curves and local-global principles.
Mercoledì 14 Novembre 2018, ore 14.30, Aula 2.
Mercoledì 14 Novembre 2018, ore 14.30, Aula 2.
Titolo: The Khukhro-Makarenko Theorem
Titolo: The Khukhro-Makarenko Theorem
Abstract: The Theorem of Khukhro and Makarenko (2007) states that if a group G admits a finite index subgroup satisfying some word laws of certain kind (called outer – or multilinear – commutators) then G admits a characteristic subgroup of finite index satisfying the same laws. Later (2015), this was extended in great generality (and in the setting of meet-semilattices) by Klyachko and Milentyeva to became a sort of principle (in the words of the authors “if somewhere there is something large and good, then there is also something large, good and symmetric”). This provides versions of the original theorem for groups, algebras, graphs and other structures. In the seminar I will illustrate these results as well as some applications.
Abstract: The Theorem of Khukhro and Makarenko (2007) states that if a group G admits a finite index subgroup satisfying some word laws of certain kind (called outer – or multilinear – commutators) then G admits a characteristic subgroup of finite index satisfying the same laws. Later (2015), this was extended in great generality (and in the setting of meet-semilattices) by Klyachko and Milentyeva to became a sort of principle (in the words of the authors “if somewhere there is something large and good, then there is also something large, good and symmetric”). This provides versions of the original theorem for groups, algebras, graphs and other structures. In the seminar I will illustrate these results as well as some applications.
Mercoledì 17 Ottobre 2018, ore 14.30, Aula 2.
Mercoledì 17 Ottobre 2018, ore 14.30, Aula 2.
Pietro Gheri (Firenze)
Pietro Gheri (Firenze)
Titolo: Subnormalizers and the degree of nilpotency of a finite group
Titolo: Subnormalizers and the degree of nilpotency of a finite group
Abstract: Let $G$ be a finite group. The degree of commutativity of $G$, $dc(G)$, is defined as the probability that two randomly chosen elements of $G$ commute. In 1973 Gustafson proved that if $G$ is non abelian then $dc(G) \leq 5/8$. As a natural generalization of this fact, one can consider the probability that two elements of $G$ generate a nilpotent subgroup, the degree of nilpotency of $G$, $dn(G)$. In 2000 Guralnick and Wilson proved a Gustafson-like result using the classification of finite simple groups: if $G$ is not nilpotent then $dn(G) \leq 1/2$. I will talk about subnormalizers (a definition due to Wielandt) and how one can use them to study the degree of nilpotency, avoiding CFSG.
Abstract: Let $G$ be a finite group. The degree of commutativity of $G$, $dc(G)$, is defined as the probability that two randomly chosen elements of $G$ commute. In 1973 Gustafson proved that if $G$ is non abelian then $dc(G) \leq 5/8$. As a natural generalization of this fact, one can consider the probability that two elements of $G$ generate a nilpotent subgroup, the degree of nilpotency of $G$, $dn(G)$. In 2000 Guralnick and Wilson proved a Gustafson-like result using the classification of finite simple groups: if $G$ is not nilpotent then $dn(G) \leq 1/2$. I will talk about subnormalizers (a definition due to Wielandt) and how one can use them to study the degree of nilpotency, avoiding CFSG.
Mercoledì 10 Ottobre 2018, ore 14.30, Aula 5.
Mercoledì 10 Ottobre 2018, ore 14.30, Aula 5.
Titolo: A reduction theorem for nonsolvable finite groups
Titolo: A reduction theorem for nonsolvable finite groups
Abstract: Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $\lambda(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called $n$-rarefied, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $\lambda(G)$ and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.
Abstract: Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $\lambda(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called $n$-rarefied, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $\lambda(G)$ and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.
Mercoledì 3 Ottobre 2018, ore 14.30, Aula 5.
Mercoledì 3 Ottobre 2018, ore 14.30, Aula 5.
Eugenio Giannelli (Firenze)
Eugenio Giannelli (Firenze)
Titolo: Problemi Local-Global per gruppi simmetrici
Titolo: Problemi Local-Global per gruppi simmetrici
Abstract: In questo seminario descriverò brevemente i concetti alla base della teoria delle rappresentazioni dei sottogruppi di Sylow P_n del gruppo simmetrico S_n. Successivamente presenterò alcuni recenti risultati (non ancora pubblicati) sulla decomposizione in costituenti irriducibili della restrizione a P_n di caratteri irriducibili di S_n. Il seminario è basato su una collaborazione con la mia studentessa di dottorato Stacey Law.
Abstract: In questo seminario descriverò brevemente i concetti alla base della teoria delle rappresentazioni dei sottogruppi di Sylow P_n del gruppo simmetrico S_n. Successivamente presenterò alcuni recenti risultati (non ancora pubblicati) sulla decomposizione in costituenti irriducibili della restrizione a P_n di caratteri irriducibili di S_n. Il seminario è basato su una collaborazione con la mia studentessa di dottorato Stacey Law.
A.A. 2017/2018
A.A. 2017/2018
Lista dei seminari (pdf).