Groups and Applications

Binary Operations: A binary operation on a set A is an everywhere defined function f: AXA→A. OR An operation which combines two elements of a set to give another element of the same set is called the binary operation. i.e. ‘*’ be a binary operation on a set A if a ∈ A, b ∈ A ⇒ a*b ∈ A (closer property) Example: Addition is a binary operation on the set N of natural numbers.

Properties of Binary operations (1) Commutative: A binary operation ‘*’ on a set A is called commutative if a *b = b*a,∀ a, b ∈ A. Example: The binary operation of addition on the set of integers Z is commutative (abelian). Associative: A binary operation * on a set A is associative if (a *b) * c = a * ( b * c ), ∀ a, b, c ∈ A . Example: The binary operation of Multiplication on the set of integers Z is Associative.

Semigroup: An algebraic structure (S, *) is called a semigroup if (i) binary operation * is closed i.e., a ∈ A, b ∈ A ⇒ a*b ∈ A (ii) Binary operation is associative. Commutative semigroup: A semigroup (S, *) is said to be a commutative if * is commutative, i.e. a *b = b*a,∀ a, b ∈ A Example (i): (N, +) is a semigroup (ii): (Z, +) is a commutative Semigroup.

Identity Element: An element e of a semigroup (S, Identity Element: An element e of a semigroup (S, *) is called an identity element if e * a = a = a * e ∀ a ∈ S. Example: (i) 0 is an identity element in (Z, +) as 0 + a = a = a + 0, ∀ a ∈ Z. (ii) 1 is an identity element in (N, .) as 1 . a = a = a . 1 , ∀ a ∈ N.

Monoid: An algebraic structure (S, Monoid: An algebraic structure (S,*) is said to be monoid if it satisfies the following properties: (i) S is closed w.r.t the operation * , i.e. a ∈ S, b ∈ S ⇒ a*b ∈ S, ∀ a, b ∈ S. (ii) Associativity: The binary operation * is associative in S if (a *b) * c = a * ( b * c ), ∀ a, b, c ∈ S. (iii) Existence of Identity element: There exiss an element e ∈ S such that a * e = a = e * a ∀ a, ∈ S In other words, a semigroup (S, *) with an identity element ‘e’ w.r.t operation * is called a monoid.

Examples (i) (N, .) is a monoid with identity element 1. (ii) (N, +) is not a monoid because for addition 0 is the identity element but 0 ∉ N.

Group Let a non empty set G together with a binary operation * defined on it is called a group if – (i) binary operation * is closed (ii) Binary operation * is associative (iii) (G, *) has an identity (iv) every element in G has inverse in G, i.e. a * b = e = b * a ,∀ a, b ∈ G.

Commutative (abelian) Group: A group (G, Commutative (abelian) Group: A group (G, *) is said to be commutative if * is commutative. Subsemigroup: Let (S, *) be a semigroup and let T be a subset of S. If T is closed under operation *, then ( T, *) is called a subsemigroup of (s, *). Submonoid: Let (S, *) be a monoid with identity e , and let T be a non empty subset of S. If T is closed under the operation * and e∈ T, then (T, *) is called a submonoid of (S, *).

Subgroup: Let (G,. ) be a group Subgroup: Let (G, *) be a group. A subset H of G is called as subgroup of G if (H, *) itself is a group. Necessary and Sufficient Condition for Subgroup: Let (G, *) be a group. A subset H of G is called as subgroup of G iff a * b-1 ∈ H, ∀ a, b ∈ H.

Cyclic Group: If every element of a group can be expressed as some powers of an element of the group, then that group is called as cyclic group. If G is a group and a is its generator, then we write G = <a>. Order of an element: The order of an element a є G is defined as the least positive integer n, if one exists, such that an = e (identity of G). Symbolically, we can write o(a) = n

Permutation Group: A permutation of degree n is a one-one mapping of a set S onto itself(i.e. bijection), where S is a finite set having n distinct elements. The number of elements in the finite set S is known as degree of permutation. Note: If S is a finite set having n distinct elements then we have n! distinct arrangements of these elements of S.

Products and Quotients of Semigroups Product of Semigroups: Let (S, *) and (T, *’) are semigroups, then the direct product of these two semigroups is the algebraic structure (S x T, *’’), in which the binary operation *’’ on S x T is defined by (s1, t1) *’’ (s2, t2) = (s1 *s2, t1 *’ t2) ∀ (s1, s2) , (t1, t2) є S x T

Homomorphism of semigroups: Let ( S,. ) and (T,. ’) be two semigroups Homomorphism of semigroups: Let ( S, *) and (T, *’) be two semigroups. An everywhere defined function f : S → T is called a homomorphism from (s, *) to (T, *’) if f(a *b) = f(a) *’ f(b) ∀ a, b є S Isomorphism of semigroups: Let ( S, *) and (T, *’) be two semigroups. An everywhere defined function f : S → T is called a Isomorphism from (s, *) to (T, *’) if (i) it is one to one correspondence from S to T & (ii) f(a *b) = f(a) *’ f(b) ∀ a, b є S .

Automorphism of semigroups: An isomorphism from a semigroup to itself is called an automorphism of the semigroup.

Cosets and Normal Subgroup Left Coset: Let (H, *) be a subgroup og ( G, *) . For any a є G, the set of aH defined by aH = { a *h/ hє H} is called the left coset of H in a determined by the element a є G. The element a is called the representative element of the left coset aH.

Right Coset: Let (H,. ) be a subgroup of ( G,. ) Right Coset: Let (H, *) be a subgroup of ( G, *) . For any a є G, the set of Ha defined by Ha = { h *a / hє H} is called the right coset of H in a determined by the element a є G. The element a is called the representative element of the right coset Ha. Normal Subgroup: If aH = Ha ⇒ H is a normal subgroup of G.