Semigroups and Monoids - Groups - Subgroups - Homomorphisms

Semigroups and Monoids – Groups - Subgroups- Homomorphisms

Definition 1: Semi-group :             [A.U N/D 2014]

A non-empty set S, together with a binary operation * is called a semi-group if * satisfies the following conditions.

(i) Closure: ∀ a, b ϵ S         ⇒ a * b ϵ S

(ii) Associative: ∀ a, b, c ϵ S, a * (b * c) = (a + b) * c

Example: (Z,.) is a semi-group.

i.e., set of integers under multiplication operation is a semi-group.

Definition 2 : Monoid :             [A.U N/D 2014]

A non-empty set M, together with a binary operation * is called a monoid if *  satisfies the following conditions

(i) Closure: ∀ a, b Є M ⇒ a * b Є M

(ii) Associative: ∀ a, b, c € M ⇒ a * (b * c) = (a *b) *c

(iii) Identity: ∀ a Є G, Ǝ e Є G

s.t. a * e = e * a = a

Example: (Z, +) is a monoid.

Definition 3: Group :

A non-empty set G, together with a binary operation * is said to form a group, if it satisfies the following conditions.

 (i) Closure: ∀ a, b Є G ⇒ a * b Є G

(ii) Associative: ∀ a, b, c Є G ⇒ a* (b*c) = (a*b) * c

(iii) Identity: ∀ a Є G, Ǝ e Є G, s.t. a * e = e * a = a

(iv) Inverse: ∀ a Є G, Ǝ a^-1 Є G, s.t. a * a^-1 = a^-1 *a = e

Example: (Z, +) is a group.

Definition 4: Abelian group:

A group (G, *) is said to be an abelian group or commutative group if a * b = b * a, ∀ a, b Є G

Example :

1. (Z, +) is an abelian group.

2. S₃ is a non-abelian group.

Definition 5: Subgroup :

A non-empty subset H of a group G (H≤G) is a subgroup of G iff a, b Є H ⇒ ab^-1Є H

Example : (Z, +) is a subgroup of group (R, +)

Definition 6: Order of a group :

Let G be a group under the binary operation *. The number of elements in G is called the order of the group G and is denoted by O (G)

Note: If the O(G) is finite, then G is called a finite group, otherwise it is called an infinite group.

Definition 7: Semi-group homomorphism

Let (S, *) and (T, ∆) be any two semigroups. A mapping g: S→T such that for any two elements a, b Є S.

g (a*b) = g (a) ∆ g (b)

is called a semigroup homomorphism.

Definition 8: Monoid homomorphism

Let (M,*, e_M) and (T, ∆, e,) be any two monoids. A mapping g: M→T such that for any two elements a, b Є M.

g (a*b) = g (a) ∆ g (b) …. (2)

g (e_M) = e_r …. (3)

is called a monoid homomorphism.

Definition 9: Group homomorphism

Let (G, *) and (H, ∆) be two groups. A mapping y: G→H is called a group homomorphism from <G, *> to <H, ∆> if for any a, b Є G.

g(a*b) = g(a) ∆ g (b)

Definition 10: Kernel of homomorphism g [ker (g)]

Let g be a group homomorphism from < G,*> to <H, ∆>. The set of elements of G which are mapped into е_H, the identity of H, is called the kernel of the homomorphism g and denoted by ker (g)

Definition 11: Cyclic group

A group G is called a cyclic group, if there exists an element a Є G such that each element of G is expressible as:

x = a_n = aa ... a (n times).

where n is some integer.

the element a Є G is called a generator of G and G is written as G = <a> or G = (a)

We also say that G is generated by a.

It may be observed that a group (G, +} is cyclic, if there exists some element a of G such that each element x of G is expressible as

x = na = a + a + ... + a (n times)

for some integer n.

Illustraion :

1. G = {-1, 1} is a cyclic group generated by -1,

since (-1)¹ = -1 = (-1)2 = 1. Thus G = (-1)

2. G (-1, 1, i, -i) is a cyclic group, where G = <i>. Notice that i¹ = i, i² = -i , i³ = -i, i⁴ = 1.

Also G = <i>.

Definition 12: Permutation :

Any one-to-one mapping of a set S onto S is called a permutation of S.

Definition 13: Even and odd permutation

A permutation of a finite set is called even if it can be written as a product of an even number of transpositions, and it is called odd if it can be written as a product of an odd number of transpositions.