Lecture 1: Binary operation, Associative operation, Groupoid, Semigroup, Monoid, Group, Four Examples of Group which includes Circle group, roots of unity.
Lecture 2A: An example of group of nth roots of unity and group of integers modulo n.
Lecture 2B: Examples: Group of prime residue classes modulo n; Group of positive real numbers with respect to multiplication; Klein Four Group; Quaternion Group; and definition of abelian group (commutative group).
Lecture 2C: Examples: General Linear Group, Special Linear Group and Orthogonal group is explained.
Lecture 2D: Examples: Symmetric group, Power set of a set with respect to symmetric difference, Dihedral Group.
Lecture 3: Some books has been suggested for reading and some problems has been given to solve.
Lecture 4A: Identity element is unique; Inverse of an element is unique; Inverse of inverse of an element; inverse of composition of two elements; Cancellation laws; . Equations Xoa=b and aoX=b, where X is unknown, are solvable in group.
Lecture 4B: Example of a groupoid where equations Xoa=b and aoX=b have unique solutions but groupoid is not group; In a groupoid if equations Xoa=b and aoX=b have unique solution then cancellation laws holds; In a finite groupoid equations Xoa=b and aoX=b have unique solution if and only if cancellation laws holds; In a semigroup if equations Xoa=b and aoX=b have solution then semigroup become group; Finite semigroup in which cancellation laws holds will become a group.
Lecture 5A: Bracket arrangement and associativity; Integral power of an element of a group; Order of an element in a group with examples.
Lecture 5B: Properties of order of an element.
Lecture 5C: Order of a Group with Examples; Example of an infinite group whose each element has finite order; Wilson Theorem.
Lecture 6A: Subgroups; Subgroup test and examples.
Lecture 6B: Examples of subgroup; Centralizer of an element and Center of the group.
Lecture 6C: Subgroup generated by subset with examples
Lecture 6D: Cyclic group and definition of Commutator subgroup of a group.
Lecture 6E: Product of subgroups; Product of two subgroups need not be subgroup; Product of two subgroups H, K of a group G is subgroup if and only if HK=KH; |HK|= |H||K|/|L| where L is intersection of H and K.
Lecture 7A: Subgroup of cyclic group is cyclic.
Lecture 7B: For each divisor d of a cyclic group of order n there is a unique subgroup of order d.
Lecture 7C: Number of elements of order $d$ in a cyclic group of order $n$ where $d$ divides $n$; Solutions of equation $x^d=1$, where one is the identity element of cyclic group of order $n$ such that $d$ divides $n$; Group theoretic proof of the formula $n=\sum_{d|n} \phi(d)$.
Lecture 8A: Left coset, Right coset with some examples.
Lecture 8B: Coset (Left or Right) need not be subgroup; For $a \in G$ , $a \in aH$, $a \in H$ if and only if $aH=H$; $G$ is contained in the union of left (right) cosets; Two cosets are either equal or disjoint.
Lecture 8C: Examples of cosets; Group and subgroup both may be infinite but number of left cosets (right costes) may be infinite or may be finite.
Lecture 8D: Bijection between $H$ and $aH$ for all $a \in G$; Bijection between the set of left cosets of $H$ in $G$ and the set of right cosets of $H$ in $G$; Lagrange's Theorem.
Lecture 8E: Applications of Lagrange's Theorem: Prime order group is cyclic; Euler Fermat Theorem and Fermat's Theorem; A group of order $p^n$ is always a $p$-group.
Lecture 9A: Normal subgroup with examples.
Lecture 9B: the Kernel of Group homomorphism; kernel as normal subgroup; $SL_n(F)$ is normal in $GL_n(F)$.
Lecture 9C: Product of two cosets of a subgroup need not be coset; Product of two cosets of a subgroup is a coset if and only if subgroup is normal; If G is a group and N is a normal subgroup, then product of coset defines a binary operation on G/N, with respect to which G/N is a group called the Factor group or Quotient group of G.
Lecture 9D: Quotient group of Klein's Four Group; Quotient of R^2 by a line passing through origin; Quotient of Quaternion group by its center; Z/nZ; Quotient of Z/20Z by 4Z/20Z.
Lecture 9E: Commutator subgroup and its properties.
Lecture 9F: Conjugate of an element; Conjugate of a subgroup; Conjugacy class of an element; Conjugacy class of subgroup; Being conjugate is an equivalence relation on elements of group; A subgroup is normal if and only if it contains conjugate of each of its elements; index of centralizer of an element is equal to the number of conjugates of that element in a finite group.
Lecture 9G: Normalizer of a subgroup; Index of the normalizer of a subgroup is equal to the number of conjugates of subgroup in a finite group.
Lecture 9H: Normal closure of a subgroup with example.
Lecture 10A: Group homomorphism with examples; Epimorphism; Monomorphism; Isomorphism; Endomorphism, Automorphism.
Lecture 10B: Composition of two group homomorphisms (if possible) is again a group homomorphism; Inverse of an isomorphism is again an isomorphism; Composition is a binary operation on End(G) and on Aut(G), where G is a group; Aut(G) is a group with respect to composition of maps called group of automorphism of group G; End(G) is semigroup with identity with respect to composition of maps.
Group Action
Group Action 1: Definition of Group action; Examples of group action; Stabilizer of a point; Stabilizer of an action,; Orbit of a point and Fixed point set of action. One can also see the same content in Group Action 1. (For Hindi see Group Action 1)
Group Action 2: Orbit, Stabilizer of point, stabilizer of action and fixed point set has been calculated for some examples of group actions.
Group Action 3: Suppose that a group G is acting on a set X. Then in this lecture it is explained that there is a subgroup H of Sym(X) such that action of G on X is same as action of H on X.
Group Action 4: Orbit stabilizer theorem; In a finite group the index of centralizer of an element is equal to the number of conjugates of that element; Index of normalizer of a subgroup in a finite group is equal to the number of conjugates of that subgroup.
Group Action 5: Class equation of an action and class equation of conjugate action has been explained with example.
Group Action 6: In a finite group of order p^n the center is always non-trivial where p is a prime; A group of prime order is cyclic; G/Z(G) is never a non-trivial cyclic group; Group of order p^2 is always abelian; The center of group of order p^3 is either of order p or equal to the group.
Group Action 7: In a finite group if p is the smallest prime divisor of order of G and G has a subgroup H of index p, then H is a normal subgroup of G.
Group Action 8: If G is a finite group and H is a proper subgroup of G such that |G| does not divide [G;H]!, then H contains a non-trivial normal subgroup of G.
Group Action 9: A problem asked in Net examination of Dec 2017 has been discussed.
Group Action 10: A problem of June 2017 of NET examination has been discussed.
Sylow Theorems
Lecture 1: Statement of Sylow theorems has been explained with example.
Lecture 2: In this lecture following points has been observed:
Observation 1: Suppose that $G$ is a group of order $p^n$. Then for each $1 \leq k \leq n$, $G$ has a subgroup of order $p^k$, ($p$ is a prime).
Observation 2: Subgroup of a subgroup is subgroup of the group.
Observation 3: If $p^k$ divides order of the group then $G$ has a subgroup of order $p^k$.
Observation 4: Sylow $p$-subgroups are $p$-groups.
Observation 5: A group $G$ has unique Sylow $p$-subgroup if and only if it is normal.
Observation 6: If $P$ is a Sylow $p$-subgroup of $G$ and $H$ is $p$-subgroup of $G$ such that $HP=PH$, then $H \leq P$. (Ref: Ramji Lal, Algebra Vol 1 Springer)
Observation 7: If $P_1$ and $P_2$ are Sylow $p$-subgroup of a finite group such that $P_1P_2=P_2P_1$, then $P_1=P_2$.
Observation 8: If $P$ is a Sylow $p$-subgroup of $G$ and $A$ is a subgroup of $G$ such that $N_G(P) \leq A \leq G$, then $A=N_G(A)$.
Observation 9: Let $M$ be a normal subgroup and $P$ be a Sylow $p$-subgroup of $M$. Then $G=M.N_G(P)$. (This is called Frattini Argument)
Observation 10: If $H$ is a normal subgroup of $p$ power order then $H$ is contained in each Sylow $p$-subgroup of the group.
Lecture 3: Proof of Sylow's first theorem using group action.
Lecture 4: Proof of Sylow's Second and Sylow's Third theorem.
Direct Product of Groups:
In the three lectures following things has been explained:
(i) External Direct Product of two groups.
(ii) Internal Direct Product of two groups.
(iii) External direct product of n copies of groups.
(iv) Internal direct product of n copies of groups.
(v) Internal direct product of countable many copies of groups.
(vi) difference between direct product and direct sum of groups
Structure theorem fot finite abelian groups: In this lecture structure theorem for finite abelian group is explained with example.