Hi!

I've the next problem:

Be G a group with neuter e. Proof the next affirmation:

If $\displaystyle a^2 = e\, \forall a\in G\, \Rightarrow \, G$ is abelian

I tried to proof the equivalent:

If $\displaystyle G$ is not abelian $\displaystyle \Rightarrow\, a^2\neq e$

Proof:

I suppose that a^2 = e

a^2 = a . a = a . a^{-1} = a^{-1} . a = e\, \mbox{ ABSURDUM G is not abelian }

This proof not convinced me so I ask for your help.

Thank's