1. ## abelian group

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 }

Thank's

2. Originally Posted by Jagger
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 }

The negation would be that if $\displaystyle G$ is no abelian there is some element of order greater than two.
All you need to note is that $\displaystyle ab=a(ab)^2b=a^2bab^2=ba$