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The concept of a binary operation on a nonempty set has already been explained in the previous. One may recall that abinary operation on a nonempty set A is just a function *:A x A → A so, for each (a, b) in A x A, * associates an element *(a, b) of A. We shall denote *(a, b) by a * b. If A is a nonempty set with a binary operation *, then A is said to be closed under *.

DEFINAITIONS AND EXAMPLES

A pair (G, *), where G is a nonempty set and * is a binary operation on G, is called a groupif the following conditions called axioms of a group, are satisfied in G:

(1) The binary operation * is associative. That is,

(a * b) * c = a *(b*c) for all a, b, c, are belong to G

(2) There is an element e in G such that

a*e = e*a = a for all a belong to G

e is called the identity element of G.

(3) For each a belong to G, there is an a.


Note:Use of word the before identity element and inverse of an element is to signify their uniqueness.




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