Question

Prove that in each group, there is only one element of identity.
That is, only one e such that e*a(where a is any element of the group) = a.

Answer 1

Start out by assuming there is a second identity, i in the group. Show that i = e.

Answer 2

Thanks.

Answer 3

Assume that there is an f such that f*a = a. e*a = a is given
Since f*a and e*a equal the same element, f*a = e*a [transitive law for equality].
Multiply both sides from the right by a^(-1) (the inverse of a):
(r*a)*a^(-1) = (e*a)*a^(-1)
f*(a*a*(-1)) = e*(a*a^(-1)) [associative property]
f = e [inverse property]
This proof hinges on the uniqueness of inverse for all elements in g.