If is a binary operation then an element a is called idempo

If * is a binary operation then an element, a, is called idempotent with respect to * if a * a = a. If * is the binary operation of a group, G, how many elements of G can be idempotent? Justify your answer.

Solution

The element is said to be idempotent if it satisfy th following property

a * a = a

but we know that there is an idenity element that satisfy

a * e = a * a

This implies that a = e, hence the element a is an identity element

Therefore, there is only one idempotent element in G