Definition of Group

An algebraic system (G,o) where G be a non-empty set with o as defined binary operation is called a group if following axiom are satisfied-

If a, b∊G ⇨ aob∊G , ∀ a,b∊G

then G is said to be closed under the binary operation.

2- Associative Law:-

If a, b, c ∊G then

(aob)oc = ao(boc) , ∀ a,b,c ∊G

If there exist e∊G such that

Aoe = eoa = a, ∀ a∊G

then e is called identity element.

If corresponding to each a∊G there exists b∊G such that

Aob = boa = e, ∀ a,b∊G then

b is called inverse of a.

A group G is called abelian if commutative law holds in it,

i.e. aob=boa, ∀ a,b∊G

The number of element in a group is called order of the group, i.e.

G={0,1,2,3}

∴ o(G)=4

Order of an element of the group:-

Let G be a group and a∊G , if there exist a least positive integer n such that

aⁿ=e (Identity element)

then n is called order of an element a.

Note:- Also o(a) ≤ o(G)