CSIR NET EXAM – GROUP THEORY NOTES

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Table of Content Group Cyclic Group Abelian Group Center of group Subgroup Conjugacy classes and conjugacy equivalence relation Homomorphism Automorphism Internal Direct Product Sylow’s Theorm

Group :

An algebraic structure (G, *), where G is a non-empty set and ‘*’ a binary operation on G is said to be a group if it is satisfies the following axioms :

(i)  a * (b * c) = (a * b) * c  for all a, b, c ∈ G

(ii)  ∃e ∈ G such that a * e = e * a = a for all a ∈ G. Here ‘e’ is called identity element of G.

(iii) for all a∈ G, ∃  b ∈ G such that a * b = b * a = e. Here b is called inverse of the element a and denoted by a^-1.

Examples of Groups :

(1) The set of integers Z is a group under addition, but not a group under multiplication.

(2) The set of rational numbers Q is a group under addition.

(3) Q* = Q/{0} is a group under multiplication, whereas Q itself is not a group under multiplication as ‘0’ does not have multiplicative inverse.

(1)      Order of a group : The number of elements in a group is called the order of the group.

(2)       Order of an element : Order of an element a Î G, is the least positive integer r such that a^r = e.

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Cyclic Group

A group (G, .) is said to be a cyclic group if there exists and element a ∈ G such that every element of G is a power of a. Element a is called a generator of G and we denote G by <a>

Examples of Cyclic Groups :

(1) The group Z of integers under addition is an infinite cyclic group generated by either 1 or -1.

(2) (Z_n, +_n) is a cyclic group of order n.

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Abelian group

A group (G, *) is said to be abelian if it satisfies commutative property, i.e. , a*b = b*a “a, b ∈ G.

Example of AbelianGroups :        

(Z, +), (Q, +), (R, +), (C, +) are all abelian groups.

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Center of group :

The center Z(G) of a group G is defined by Z(G) = {z ∈ G / zx = xz for all x ∈ G}.

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Subgroup

A non-empty subset H of G is said to be a subgroup of G if H itself is a group under same binary composition as that of G.

Example :

(1) (Z, +) is subgroup of (Q, +).

(2) Z(G) is a subgroup of G.

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Normal Subgroup

A subgroup H of a group (G, .)is said to be a normal subgroup of G is Ha = aH for all  a∈G and denoted by HΔG .

Important Results on Normal Subgroups :

(1) A subgroup H of a group G is normal if and only if g^-1hg ∈ H for every h ∈ H, g ∈ G.

(2) Z(G) is a normal subgroup of G.

(3) A subgroup of index 2 in a group G is a normal subgroup of G, i.e., H £ G and [G : H] = 2, then H is normal subgroup of G.

Simple Group :

A group G ¹ {e}, which does not have any proper normal subgroup is called a simple group.

Example :

Z_p is a simple group if and only if p is prime.

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Quotient Group :

Let (G, .)be a group and N be a normal subgroup of G then we define

Example – 1 : Q/Z is a quotient group

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Conjugacy classes and conjugacy equivalence relation

1. Conjugate elements : Let G be a group and a, b ∈ G.a is said to be a conjugate of b in G if there eixts c∈G such that a = c^-1bc. We say a is related to b iff a is a conjugate of b.
2. Conjugacy classes :Let a ∈ G, then collection of all conjugate elements of a is congugacy class of a and denoted by Cl(a).

Class equation

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Homomorphism:

Let G and G’ be any two groups and let ‘o’ and ‘*’ denote their respective binary operations. Then a mapping f : G → G’ is called a homomorphism, if f(a  ° b) = f(a) * f(b) ” a, b ∈ G.

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Automorphism:

An isomorphism of a group G onto itself is called an automorphism of G.

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Inner automorphism:

Let a be an element of a group G. The automorphism f_a : G → G given by f_a(x) = axa^-1, x∈G is called an inner automorphism of G determined by a. Inn(G) denotes the set of all inner automorphisms of G.

Note : Set of all automorphisms from G to G form a group under composition and is denoted by Aut (G).

Kernel of a Homomorphism:

Let f be a homomorphism of a group G into a group G’ then the kernel (f)={x ∈ G | f(x) = e’}.

where e’ is the identity of G’ We denote the kernel (f) by ker(f).

Note : Let f : G → G’ then ker f is a normal subgroup of G.

Internal Direct Product :

A group G is an internal direct product of its subgroups H and K if and only if

(i) elements of H and K commute.

(ii) every element of G can be expressed uniquely in the form hk, where h∈H and k∈K.

SYLOW’S THEOREM

p-subgroup :

Let p be a prime number. Then a subgroup H of a group G is called a p-subgroup if the order of each element of H is a power of p.

Sylow’s First Theorem :

Let G be infinite group of order n = p^kq(k≥1), where p & q are prime number such that gcd(p, q) = 1. Then for each i, (1 ≤  i ≤ k), G has at least one subgroup of order p

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Cauchy’s Theorem for finite groups :

If a prime number p divides the order of a finite group G, then G contains at least one element of order p.

Sylow’s Second Theorem :

Let G be a finite group and p be a prime number such that p | O(G). Then all Sylow p-subgroup of G are conjugates of one another.

Sylow’s Third Theorem :

Let G be a finite group and p be a prime number such that p | O(G). Then the number of Sylow p-subgroups is of the form 1+mp, where m is some non-negative integer such that (l + mp) | O(G).

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