1. ## Groups problem...(Axioms)

Say S is a set of elements (finitely or infinetly). Suppose that S is closed under the operation *, and that the associative law holds. If a*e=a and there exists -a (the inverse element) such that a*(-a)=e for all a in S. Then e*a=a and (-a)*a=e and the uniqueness of e (the identity element) can be deducted.

A little help with this is needed. Thanks in advance.

2. Originally Posted by Kichigo
Say S is a set of elements (finitely or infinetly). Suppose that S is closed under the operation *, and that the associative law holds. If a*e=a and there exists -a (the inverse element) such that a*(-a)=e for all a in S. Then e*a=a and (-a)*a=e and the uniqueness of e (the identity element) can be deducted.

A little help with this is needed. Thanks in advance.

First prove that in $\displaystyle S\,,\,\,x\cdot x=x\Longrightarrow x=e$ , and then prove that $\displaystyle (a\cdot a^{-1})(a\cdot a^{-1})=\a\cdot a^{-1}$ (it seems to be that $\displaystyle a^{-1}$ is more usual and clearer to denote the inverse of $\displaystyle a$ than $\displaystyle -a$).
Finally, use the above to prove $\displaystyle e\cdot a=a$

Tonio