Group axioms proof

Question

A group is a collection G that satisfies the following 3 axioms:

1) associativity a·(b·c)= (a·b)·c
2) identity element e: e·a=a ∀a in G
3) Inverse. Each element a in G has an inverse a^-1 such that a^-1·a=e

I want to prove that these three are sufficient to get a·e=a and a·a^-1=e but insufficient to get a·a^-1=e alone

If anyone can help me in steps that would be appreciated :)

Answer 1

This should help:

 

http://math.stackexchange.com/questions/537572/any-set-with-associativity-left-identity-left-inverse-is-a-group

 

You need only read the first three paragraphs (until e = aa^-1). If there's anything there that doesn't make sense, leave a comment and I'll clarify the person's work.