Monoids and Groups

Binary Operation: - Let S be a non-empty set, and * said to be a binary operation on S if a * b is defined for all a, b ∊ S. In other words, * is a rule for any two elements in the set S.

Closure Property: - Let S be a non-empty set, A binary operation * on S is said to be a closed binary operation on S, if 

a * b ∊ S, for all a, b ∊ S.

Associative Property: - Let S be a subset of Z. A binary operation * on S is said to be associative, if 

(a * b) * c = a * (b * c) for all a, b, c ∊ S

Commutative Property: - Let S be a non-empty set. A binary operation * on S is said to be Commutative, if 

a * b = b * a for all a, b ∊ S.

Identity Property: - A non-empty set S with binary operation *, is said to have an identity e ∊ S, if

e * a = a * e = a, for all a ∊ S

Distributive Property: - Let S be a non-empty set. Let *₁ and *₂ be two different binary operations on S. 

Then *₁ is said to be distributive over *₂ on S if a *₁ (b *₂ C) = (a *₁ b) *₂ (a *₁ c), for all a, b, c ∊ S.

Inverse Property: - Let S be a set with a binary operation *, and with identity e. Let a ∊ S, then b ∊ S is called an inverse of a if 

a * b = b * a = e. (a * a’ = a’ * a = e)

Cancellation: - Let S be a non-empty set and a binary operation * on A. Then the operation * has the cancellation property, if for every a, b, c ∊ A we have ….

A * b = a * c => b = c   {Left concellation}

B * a = c * a => b = c   {Right concellation}

Semi Group: - Let us consider, an algebraic system (A, *), where * is a binary operation on A. Then, the system (A, *) is said to be semi-group if satisfies the following properties: -

• Closed Property
• Associative Property

Sub-Semi Group: - Consider a semigroup (A, *) and let B ⊆ A. Then the system (B, *) is called a sub-semigroup if the set B is closed under the operation *.

Monoid: - Let us consider an algebraic system (A, *), where * is a binary operation on A. Then the system (A, *) is said to be a monoid if it satisfies the flowing properties.

• Closed Property
• Associative Property
• Identity Property

Sub Monoid: - Let us consider a monoid (M, *) also let S ⊆ M. Then (S, *) is called sub-monoid of (M, *) if and only if it satisfies the following properties.

• S is Closed
• Identity

Group: - A non-empty set G (G, *) is called a group and A group is always a monoid, semigroup, and algebraic structure if it follows properties.

• Closure
• Associativity
• Identity Element
• Inverses

Abelian Group: - A non-empty set S (S, *) is called an abelian group if it follows the following properties.

• Closure
• Associativity
• Identity
• Inverse
• Commutative

Finite Group: - A Group (G, *) is called a finite group if G is a finite set.

Infinite Group: - A Group (G, *) is called an infinite group if G is an infinite set

Order of Group: - The order of group G is the number of elements in group G. it is denoted by |G|. A group of order 1 has only the identity element, ({e} *).

Sub Group: - A Sub-Group H is a subset of a group G if it satisfies the 4 Group Properties.

• Closure
• Associativity
• Identity Element
• Inverses

Cyclic subgroup: - A subgroup K of a group G is said to be a cyclic subgroup if there exists an element x ∊ G such that every element of K can be written in the form xⁿ for some n ∊ Z

Coset: - Let H be a subgroup of a group G and a ∊ G, then the set

aH = {ah | h ∊ H} is called a left coset of H in G.

Ha = {ha | h ∊ H} is called a right coset of H in G

By this definition, it is clear that corresponding to every element of G, we have a left coset and right coset of H in G. it obvious that

aH ⊂ G, Ha ⊂ G

Homomorphism (Group Morphism): - A mapping f from a group (G, *) to a group (G’, o) is called a group homomorphism from G to G’ if

                f(a * b) = f(a) o f(b) for all a, b ∊ G

        Thus, if f is a morphism from G to G’, then it preserves the composition in both the groups G and G’ i.e.

        Image of the composite = Composite of image

Theorem 1. – Iff is a homomorphism from a group G to G’ and if e and e’ be their respective identities, then:

            1. f(e) = e’

            2. f(a^-1) = [f(a)]^-1 ,   a ∊ G

    Prove: -

1)                    Let a ∊ G, then ae = a = ea

                ⇒ f(ae) = f(a) = f(ea)

                ⇒ f(a) f(e) = f(a) = f(e) f(a)

                ⇒ f(e) is the identity in G’ 

                ⇒ f(e) = e’

        Therefore, the image of the identity of G under the group morphism f is the identity of G’

2)                  Let a^-1 be the inverse of a ∊ G, then aa^-1 = e = a^-1 a

                ⇒ f(aa^-1) = f(e) = f(a^-1 a)

                ⇒ f(a) f(a^-1) = e’ = f(a^-1)

                ⇒ f(a^-1) = [f(a)]^-1

        Therefore, the f-image of the inverse of any element of G under f is the inverse of the f-image of a in G’

Kernel of Homomorphism: - Let f be a homomorphism of a group G into G’, then the set K of all those elements of G Which are mapped to the identity e’ of G’ is called the kernel of the homomorphism f.

It denoted by ker f or ker(f)

Ker(f) = {x ∊ G | f(x) = e’}

Isomorphism: - A morphism f of a group (G, *) to a group (G’, o) is an isomorphism if

                1.    f is one – one {f(a) = f(b) => a = b}

                2.    f is onto {f(G) = G’}

                3.    f is morphism {f(a*b) = f(a) o f(b)}

            From the above definition, it is clear that a group morphism is an isomorphism iff is a bijection.

Auto morphism: - An isomorphic mapping of a group G onto G itself is called an automorphism of G

It is denoted by A_G or Aut(G)

A mapping f: G -> G is called an automorphism of G is

                1.    f is a bijection

                2.    f(ab) = f(a) f(b)