Groups, Subgroups, Cyclic Groups and Permutation Groups - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

Introduction - Groups

Definition -  A group (G,·) is a nonempty set G together with a binary operation · on G such that the following conditions hold:

(i) Closure: For all a,b  G the element a · b is a uniquely defined element of G.

(ii) Associativity: For all a,b,c  ∈ G, we have

a · (b · c) = (a · b) · c.

(iii) Identity: There exists an identity element e ∈ G such that:

e · a = a and a · e = a, for all a ∈ G.

(iv) Inverses: For each a  G there exists an inverse element a^-1 ∈ G such that:

a · a^-1 = e and a^-1 · a = e.

We will usually simply write ab for the product a · b.

Proposition 1 (Cancellation Property for Groups): Let G be a group, and let a,b,c ∈ G.

(a) If ab=ac, then b=c.

(b) If ac=bc, then a=b.
Definitions

• Abelian Group -  Group G is said to be abelian if ab=ba for all a,b ∈ G.

• Finite Group and order of group - Group G is said to be a finite group if set G has a finite number of elements. In this case, the number of elements is called the order of G, denoted by |G|. Let a be an element of the group G. If there exists a positive integer n such that aⁿ = e, then a is said to have finite order, and the smallest such positive integer is called the order of a, denoted by o(a).
• Infinite Group - If there does not exist a positive integer n such that aⁿ = e, then a is said to have infinite order.
• Subgroup - Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G.

Proposition 2: Let G be a group with identity element e, and let H be a subset of G. Then H is a subgroup of G if and only if the following conditions hold:

1. ab ∈ H for all a,b ∈ H;
2.  e ∈ H;
3.  a^-1 ∈ H for all a ∈ H.

Theorem (Lagrange): If H is a subgroup of the finite group G, then the order of H is a divisor of the order of G.

Corollary 1: Let G be a finite group of order n.

1. For any a ∈ G, o(a) is a divisor of n.
2. For any a ∈ G, aⁿ = e.

Example (Euler's theorem): Let G be the multiplicative group of congruence classes modulo n. The order of G is given by (n), and so by Corollary 1, raising any congruence class to the power (n) must give the identity element.

Corollary 2: Any group of prime order is cyclic.

Isomorphism: Let G₁ and G₂ be groups, and let θ: G₁ -> G₂ be a function. Then  is said to be a group isomorphism if

1.  θ is one-to-one and onto and
2.  θ(ab) = θ(a) θ(b) for all a,b ∈ G₁.

In this case, G₁ is said to be isomorphic to G₂, and this is denoted by G₁ ≌ G₂.

Proposition 3: Let θ: G₁ -> G₂ be an isomorphism of groups.

1. If a has order n in G₁, then θ(a) has order n in G₂.
2. If G₁ is abelian, then so is G₂.
3. If G₁ is cyclic, then so is G₂.

Cyclic groups

Let G be a group, and let a be any element of G. The set is called the cyclic subgroup generated by a. <a> = {x ∈ G | x = aⁿ for some n ∈ Z}

Group G is called a cyclic group if there exists an element a  G such that G=<a>. In this case, a is called a generator of G.

Proposition 4: Let G be a group, and let a ∈ G.

1. The set <a> is a subgroup of G.
2. If K is any subgroup of G such that a  K, then <a> ⊂ K.

Proposition 5: Let a be an element of group G.

1. If a has infinite order, and a^k = am for integers k,m, them k=m.
2. If a has finite order and k is any integer, then a^k = e if and only if o(a) | k.
3. If a has finite order o(a)=n, then for all integers k, m, we have a^k = a^m if and only if k ≡ m (mod n). Furthermore, |<a>|=o(a).

Corollaries to Lagrange's Theorem (restated):

(a) For any a ∈ G, o(a) is a divisor of |G|.

(b) For any a ∈ G, aⁿ = e, for n = |G|.

(c) Any group of prime order is cyclic.

Theorem: Every subgroup of a cyclic group is cyclic.

Theorem: Let G cyclic group.

1. If G is infinite, then G ≌ Z.
2. If |G| = n, then G ≌ Z_n.

Proposition 6: Let G = <a> be a cyclic group with |G| = n.

(a) If m  Z, then <a^m> = <a^d>, where d=gcd(m,n), and am has order n/d.

(b) The element a^k generates G if and only if gcd(k,n)=1.

(c) The subgroups of G are in one-to-one correspondence with the positive divisors of n.

(d) If m and k are divisors of n, then <a^m> ⊂ <a^k> if and only if k | m.

Exponent of G: Let G be a group. If there exists a positive integer N such that a^N= e for all a ∈ G, then the smallest such positive integer is called the exponent of G.

Lemma: Let G be a group, and let a,b ∈ G be elements such that ab = ba. If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).

Proposition 7: Let G be a finite abelian group.

(a) The exponent of G is equal to the order of any element of G of maximal order.

(b) Group G is cyclic if and only if its exponent is equal to its order.

Permutation groups

A permutation group is a finite group G  whose elements are permutations of a given set and whose group operation is composition of permutations in G.  Permutation groups have orders dividing n!.

The set of all permutations of a set S is denoted by Sym(S). The set of all permutations of the set {1,2,...,n} is denoted by S_n.

Proposition 8: If S is any nonempty set, then Sym(S) is a group under the operation of the composition of functions.

Theorem: Every permutation in S_n can be written as a product of disjoint cycles. The cycles that appear in the product are unique.

Proposition 9: If a permutation in S_n is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.

Theorem: (Cayley) Every group is isomorphic to a permutation group.

Dihedral Group: Let n > 2 be an integer. The group of rigid motions of a regular n-gon is called the nth dihedral group, denoted by Dn.

We can describe the nth dihedral group as

D_n = {a^k, a^kb | 0 < k < n},

subject to the relations o(a) = n, o(b) = 2, and ba = a^-1b.

Theorem: If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.

Even and Odd Permutation: A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.

Proposition 10: The set of all even permutations of S_n is a subgroup of S_n.

Alternating Group: The set of all even permutations of S_n is called the alternating group on n elements and will be denoted by A_n.

Other examples

(Group of units modulo n) Let n be a positive integer. The set of units modulo n, denoted by Z_n^×, is an abelian group under the multiplication of congruence classes. Its order is given by the value φ(n) of Euler's phi-function.

General Linear Group. The set of all invertible n × n matrices with entries in R is called the general linear group of degree n over the real numbers and is denoted by GL_n(R).

Proposition. The set GL_n(R) forms a group under matrix multiplication.

Definition. Let G₁ and G₂ be groups. The set of all ordered pairs (x₁,x₂) such that x₁ ∈ G₁ and x₂ ∈ G₂ is called the direct product of G₁ and G₂, denoted by G₁ × G₂.

Proposition. Let G₁ and G₂ be groups.

1. The direct product G₁ × G₂ is a group under the multiplication defined for all
   (a₁,a₂), (b₁,b₂) ∈ G₁ × G₂ by
   (a₁,a₂) (b₁,b₂) = (a₁b₁,a₂b₂)
2. If the elements a₁ ∈ G₁ and a₂ ∈ G₂ have orders n and m, respectively, then in G₁ × G₂ the element (a₁,a₂) has order lcm[n,m].

Field: Let F be a set with two binary operations + and · with respective identity elements 0 and 1, where 1 is distinct from 0. Then F is called a field if

(i) the set of all elements of F is an abelian group under +;

(ii) the set of all nonzero elements of F is an abelian group under ·;

(iii) a · (b+c) = a · b + a · c for all a,b,c in F.

General Linear Group of Degree n: Let F be a field. The set of all invertible n × n matrices with entries in F is called the general linear group of degree n over F, and is denoted by GL_n(F).

Proposition. Let F be a field. Then GL_n(F) is a group under matrix multiplication.

Proposition. If m,n are positive integers such that gcd(m,n)=1, then

Example. 3.3.7. (Quaternion group)
Consider the following set of invertible 2 × 2 matrices with entries in the field of complex numbers.

If we let

then we have the identities

i² = j² = k² = -1;

ij = k, jk = i, ki = j;

ji = -k, kj = -i, ik = -j.

These elements form a nonabelian group Q of order 8 called the quaternion group, or group of quaternion units.