MATH 603: Topics in Group Theory

Fall 2019

New: Solutions to selected exercises in “Finite Groups: An Introduction by Jean-Pierre Serre" PDF. (Last update: December, 2019)

• Main reference: Serre, J.P., 2016. Finite groups: An Introduction.

• Schedule: MWF 1:10 pm - 2:10 pm; Rm: WH-309.

• Office hours: MWF 2:30 pm - 3:30 pm in WH-107.

• This course will give graduate students a solid introduction to some fundamental topics in group theory based on Serre's book on Finite Groups. It will be complemented with topics on infinite groups towards the end. The goal is to prepare graduate students to work with groups. A summary of the announcements/discussion being posted in MyCourses is available below.

• Faculty research in Group Theory and related areas at Binghamton University.

• Comments on Serre, Finite groups: an introduction by B. Poonen et al.

Week 1:

Lecture 1: Discussion of finite groups of order up to 12, up to isomorphism; action of dihedral groups on the plane to motivate group actions; G-sets; transitive actions, faithful actions, free actions; orbits, stabilizer subgroups; Examples: action of the symmetric group on n-letters on an n-element set; left multiplication action, conjugation action; G-torsors.

Lecture 2: Part 1: More examples: action of GL(2,C) on the complex projective line; the orbit-stabilizer theorem; Burnside's lemma.

Part 2: normal subgroups, short exact sequences involving Aut(G), Int(G), Out(G); simple groups, examples of finite simple groups.

Homework 1: Section 1.5: 1-8 and 2 more exercises of your choice not selected by other students.

Week 2:

Lecture 3: General construction of normal subgroups such as characteristic subgroups which include the center of a group and the derived subgroup; filtratons, Jordan-Holder series/composition series, the Jordan-Holder theorem; Proof that every finite group has a Jordan-Holder filtration, examples of composition series, an application of Jordan-Holder theorem: uniqueness of prime factorization of integers.

Suggested presentations: The families of finite groups whose simplicity you can show in class; Subgroups of products: Goursat's and Ribet's lemmas.

Lecture 4: Proof of Lagrange's theorem; p-groups, p-Sylow subgroups, examples of abelian and nonabelian p-Sylow subgroups; Proof of Sylow's first theorem on existence of p-Sylows.

Lecture 5: Sylow's second theorem; properties of p-Sylow subgroups of normal subgroups and quotients, a factorization of a finite group via Frattini's argument.

Week 3:

Lecture 6: More properties of p-Sylow subgroups; a second proof of Sylow's First theorem via embeddings; Fusion in the normalizer of a p-Sylow subgroup with examples.

Lecture 7: Local conjugacy, examples in GL(3, Z/pZ), Alperin's theorem; construction of a p-Sylow subgroup of symmetric groups via automorphisms of a rooted p-ary tree.

Week 4:

Lecture 8: Commutator (derived) subgroups, abelianization of a group, examples: abelian groups, free groups, symmetric groups, the quaternion group, GL(n,K); definition of solvable groups, motivation from Galois theory; equivalent characterization of solvable groups.

Homework 2: Section 2.7: 1-5 and 2 more exercises of your choice not selected by other students.

Lecture 9: Examples of solvable groups of derived length 2,3 and 4, general examples: Burside's p^aq^b theorem; Feit-Thompson odd order theorem; proof that solvability is closed under taking subgroups, quotients, and finite direct products; characterization of a finite solvable group via quotients of its Jordan-Hölder series; the first (in group order) few nonsolvable finite groups: A_5, S_5, SL(2,5), A_5xZ/2, PSL(3,2); nilpotent groups, examples: the quaternion group Q_8, the discrete Heisenberg groups H_3(Z) and H_3(Z/n).

Lecture 10: Hints for exercise 2.7 4(c) on automorphisms of the nonabelian group of order pq and proof that S_n and A_n are not solvable for n>=5; nilpotent implies solvable; characterizing nilpotency via central extensions; existence of nontrivial centers in nontrivial nilpotent groups; characterizing nilpotency via central filtrations.

Week 5:

Lecture 11: Lie algebras refresher: definition of a Lie algebra over a field, Lie brackets; examples: trivial examples: abelian Lie algebras, nontrivial examples: three dimensional vectors with cross-product, gl(n,k), sl(n,k); the universal enveloping algebra of a Lie algebra, dimension, ideals, simple ideals, Lie subalgebras; classification of Lie algebras of small dimension up to isomorphism; solvable Lie algebras, nilpotent Lie algebras; Lie algebras associated to a central filtration.

Lecture 12: Lie algebra associated to a nilpotent group; an instance of the use of Lie algebras: proving a homomorphism of a group H into a nilpotent group G which is surjective to G^ab is surjective to G ; examples of nilpotent subgroups of GL(V): a Borel subgroup constructed from a complete flag of a vector space V; upper unitriangular matrix groups have nilpotence class n-1.

Lecture 13: Presentations on homework exercises.

Week 6:

Lecture 14: Presentations on homework exercises.

Lecture 15: Presentations on homework exercises:

Garrett Proffitt: On the simplicity of A_5: Following Serre's recommendation, we prove the simplicity of A_5 by counting conjugacy classes. For greater generality, we also develop a lemma on how conjugacy classes of S_n may split when we descend to A_n and reprove a standard lemma that normal subgroups are unions of conjugacy classes. The result follows almost immediately from the two lemmas.

Lecture 16: Discussion of hints for exercises in Chapter 2; remaining presentations on homework exercises.

Week 7:

Lecture 17: Proof every p-group is nilpotent; proofs of properties of finite p-groups: proper subgroups are not self-normalizing, subgroups of index p are normal, proper subgroups are contained in index p subgroups, nontrivial p-group has nontrivial center; proof of statements equivalent to nilpotency of a finite group: product of p-groups for some finite list of primes p, unique p-Sylow subgroup for each p, any p-Sylow and q-Sylow subgroups centralize each other, any two elements of relatively prime orders commute.

Lecture 18: More discussion on statements equivalent to nilpotency of a finite group from previous class; the structure theorem of finitely generated modules over a PID with examples listing all finite abelian groups of given order by invariant factors and elementary divisors.

Week 8:

Lecture 19: The Frattini subgroup of a finite group as the intersection of all maximal subgroups, examples; proofs of its properties: is a characteristic subgroup, is the set of all non-generator elements, is a nilpotent subgroup, has the form (G,G)G^p when G is a finite p-group; applications to rank and automorphism group of a finite p-group.

Homework 3: Section 3.11: 1-3 and 2 more exercises of your choice not selected by other students.

Lecture 20: G-modules generalize representations of a group G; taking G-invariants is not necessarily right exact which motivates introduction of so-called cohomology groups; n-cochains, the coboundary map d, the basic formula (d^2=0); n-cocycles, n-coboundaries; definition of n-th cohomology group of G with values in A where A is a G-module; low-dimensional examples.

Week 9:

Lecture 21: Characterizations of 0-th, first and second cohomologies: 0-th cohomology group in terms of fixed elements, 1st cohomology group in terms of crossed homomorphisms and Hom(G, A) when G acts trivially on A, 2nd cohomology group in terms of factors sets and normalized factor sets; vanishing criteria for cohomology of finite groups.

Lecture 22: An extension of a group by another group; equivalent (isomorphic) extensions, Examples: the 4 nonisomorphic extensions of the Klein four group by the cyclic group Z/2; a section of an extension, a lifting of a group in its extension, splitting extensions or equivalently (external or internal) semidirect products.

Lecture 23: Examples of nonsplit extensions: The quaternions Q8 as an extension of Z/2 by the Klein four group, p-primary cyclic groups as nontrivial extensions of Z/p; complement of a normal subgroup: e.g. complements of A3 in S3, uniqueness of complements up to isomorphism; semidirect products in terms of complements; Examples of semidirect products: Sn as An by Z/2, the dihedral group D2n as Z/n by Z/2, the holomorph of a group G, i.e. Hol(G) as G by Aut(G); nonexamples: p-primary cyclic groups are not a semidirect product of any two nontrivial groups; dependence of semidirect product on how the group splits over the normal subgroup: e.g S3 and Z/6 are both semidirect products of Z/3 by Z/2 but are nonisomorphic as they split differently over Z/3.

Week 10:

Lecture 24: Extensions with abelian kernel: proof that the isomorphisms of extensions of G by the G-module A (abelian) are in one-to-one correspondence with the elements of the cohomology group H^2(G,A) as follows. Split extensions correspond to the trivial element in H^2(G,A). A corollary is that when G and the G-module A are finite of coprime orders, every extension of G by A splits.

Lecture 25: Homework break.

Lecture 26: Interpreting H^1(G,A) in terms of splitting extensions: proof that the conjugacy classes (by elements of A) of splittings of E are in bijective correspondence with the elements of the cohomology group H^1(G,A). Two corollaries are (i) the splittings of a split extension are conjugate to each other if and only if H^1(G,A) vanishes; (ii) if G and A are finite of coprime orders and E is an extension of G by A, then every two splittings of E are conjugate.

Week 11:

Lecture 27: Extensions with arbitrary kernel (A not necessarily abelian): Given an extension E of G by A, we can construct a homomorphism psi:G -> Out(A) that results in psi_0:G->Aut(Z(A)) making Z(A) into a G-module. In general, given a homomorphism psi: G->Out(Z(A)), associate to psi an element c(psi) of the cohomology group H^3(G, Z(A)); sketch a proof of a theorem of Eilenberg and Mac Lane: there exists an extension of G by A corresponding to psi if and only if the 3-cocycle c(psi) is trivial. The obstruction to lifting psi: G->Out(A) to G->Aut(A) is determined by nontriviality of the 3-cocycle c(psi).

Lecture 28: Given a homomorphism psi:G -> Out(A), construction of an action of the cohomology group H^2(G,Z(A)) on the set Ext(G,A,psi), the isomorphism classes of extensions of G by A corresponding to psi. Discussion of the theorem that (i) the action above is free, and (ii) the action is transitive if Ext(G,A,psi) is nonempty. Corollaries are (i) if G and Z(A) are finite, so is Ext(G,A,psi) and the cardinality of Ext(G,A,psi) is either 0 or the order of H^2(G,Z(A)), (ii) the cardinality of Ext(G,A,psi) is 1 if both H^2(G,Z(A)) and H^3(G,Z(A)) are trivial. Application: when Z(A)=1, there is only one extension (upto isomorphism) corresponding to psi:G -> Out(A). Examples of centerless finite groups: nonabelian finite simple groups, the dihedral groups D_{2n} with n odd, the alternating group A_4, the symmetric groups Sn with n>2. Statement of Schur-Zassenhaus theorem; motivation for defining Hall subgroups.

Lecture 29: Summary of the use of cohomology groups to study group extensions. Corollaries of Schur-Zassenhaus theorem: e.g. the order of a p-sylow subgroup and its index in its normalizer are coprime. This gives rise to a split extension. A group whose composition series is a series of normal Hall subgroups is an iterated semidirect product of its composition factors; a partial solution to the group extension problem.

Week 12:

Lecture 30: Liftings of homomorphisms to group extensions; pullback of an extension over a homomorphism; consequences of Schur-Zassenhaus theorem to liftings of homomorphisms. Applications to p-adic liftings: construction of p-adic numbers via the algebraic approach (projective/inverse limits) and via the analytic approach (metric completion), basic ring properties of the p-adic numbers.

Lecture 31: An example of lifting of a group representation from characteristic p to characteristic 0. Wreath products: unrestricted, restricted and regular wreath products; examples: the lamplighter group, generalized symmetric group, a sylow p-subgroups of the symmetric group on p^n letters is isomorphic to the iterated regular wreath product of n copies of the cyclic group of order p.

Lecture 32: Presentations on homework exercises: Jonathan Doane, Michael Gottstein (Iwasawa's simplicity criterion), Sayak Sengupta.

Week 13:

Lecture 33: pi-groups; Hall pi-subgroups; Hall's theorem: a generalization of Sylow's theorems for finite solvable groups; consequences: e.g. two pi-Sylow subgroups of a solvable group are conjugate; Theorem: If for every set pi of prime numbers, G contains a pi-Sylow subgroup, then G is solvable; Nonexamples: the alternating group A_4 has order 12 and 6 divides 12 but A_4 has no subgroups of order 6. So, Hall's theorem does not extend to all divisors of the order a finite solvable group; the group A_5 is not solvable and its order 60 has Hall divisors 10 and 15 but A_5 has no subgroups of these orders. So, Hall's theorem does not extend to nonsolvable groups. Permutable subgroups; Sylow systems (permutable families of Sylow subgroups); a more precise version of Hall's theorem: i) any solvable group has a Sylow system. ii) any two Sylow systems are conjugate. Sylow-like properties of pi-subgroups.

Lecture 34: The size of the union of conjugates of a subgroup of a finite group; Jordan's theorem : A group acting transitively on a finite set of size >1 has an element that does not fix any element of the set; a Frobenius pair, Frobenius groups, Frobenius theorem: given a Frobenius pair (G,H), the identity element together with the elements of G which don't belong to any conjugate of H, make up a normal subgroup N of G such that N intersects H trivially and G =NH. The Frobenius kernel of a group and its uniqueness, Frobenius complement. Examples: G=N.C2 with N an abelian group of odd order and C2, the group of order 2, acting on N by inversion; nonabelian groups of order pq (the group of invertible affine linear maps on the affine line over a finite field). Every Frobenius group has a trivial center, so we get lots of nonexamples: nontrivial nilpotent groups (e.g p-groups).

Lecture 35:

Part I: Garrett Proffitt: All finite groups are involved in the mapping class group: We provide an outline of Masbaum and Reid's paper of the same name and prove that for fixed genus g, every finite group occurs as a quotient of a finite index subgroup of the mapping class group Mod(S_g). While the bulk of the paper is beyond the scope of our group theory class, we discuss the notion of a mapping class group, prove Mod(T^2)=SL2(Z) for the torus, and finally prove the lemma that links the secondary theorem of the paper to the main theorem. PDF summary.

Part II: Zachary Costanzo: A finite group is nilpotent if and only if it is p-nilpotent for every p.

Week 14:

Lecture 36: Definition of the transfer map Ver: G^ab-->H^ab for a subgroup H of finite index in a group G (not necessarily finite); the map is a homomorphism and does not depend on the choice of section phi:G/H-->G; proof that the composite map G^ab-->H^ab -->G^ab is g-->g^n where |G/H| = n. Corollary: If G is abelian, then Ver(g)=g^n for every g in G. Examples of transfer: Gauss lemma and an example computation of the Legendre symbol (2/p) for p odd prime, relation to power residue symbols; (to be covered next time) application to infinite groups: If G is a torsion-free group with a finite index subgroup H isomorphic to the integers Z, then G is isomorphic to Z.

Lecture 37:

Jonathan Doane: Basics of character theory of groups: definition of linear representation of a group on a vector space; the group algebra point of view of linear representations; examples: direct sums, tensor products, dual, exterior and symmetric powers of representations of a group; character of a representation, properties of characters, examples: permutation characters, the character of the left regular representation.

Lecture 38:

Part I: Andrew Lamoureux: Discussion of results used in lifting a representation from characteristic 0 to characteristic p: the construction and basic properties of the p-adic numbers and p-adic integers, such as their arithmetic and their topology.

Part II: Basics properties of characters of finite groups; existence of a G-invariant positive definite hermitian form on a complex vector space for G a finite group; splitting of linear representations.

Week 15:

Lecture 39: Proof of theorem: every representation is a direct sum of irreducible representations; Schur's lemma, orthogonality of irreducible characters, proof of corollary: two representations with the same character are isomorphic.

Week 16:

Lecture 40: Cancelled (Snow day.) Reading assignment: Minkowski's theorem on the finite subgroups of GLn(Q).

Lecture 41:

Part I: Garrett Proffitt: Jordan's theorem on the finite subgroups of GLn(C): Following Serre's exposition, we outline the proof that for every n>=1, there exists a real number f(n) such that every finite subgroup of GLn(C) contains an abelian normal subgroup of index less or equal to f(n). In fact, we establish an explicit formula for f(n) originally given by Frobenius in 1968. The proof is an entertaining adventure through linear algebra over C, nuances from functional analysis, and a geometric argument about well-spaced points on a sphere.

Part II: Light introduction to computational software for studying finite groups: GAP, MAGMA and SAGE; example codes.