Conjugacy class enough to make it Abelian?

Question

I am asked to prove:

Any group containing only second order elements and identity is Abelian.

Is it enough to say that because each element is a conjugacy class by itself, then the group has to be Abelian?

Thanks.

One usually says *elements of order $2$*. – Andreas Caranti Dec 27 '16 at 17:40 — Andreas Caranti, Dec 27 '16 at 17:40
if every element of order 2 then $a^2=e$ then $a=a^{-1}$ – Mustafa Dec 27 '16 at 17:43 — Mustafa, Dec 27 '16 at 17:43

score 2 · Accepted Answer · answered Dec 27 '16 at 17:28

2

$(ab)^2=e=a^2b^2$ for all $a,b\in G$. Can you take it from here?

answered Dec 27 '16 at 17:28

Asinomás

score 0 · Answer 2 · answered Dec 27 '16 at 17:47

0

Let $x,y\not=e$ and $x\not=y^{-1}.$ Then $$xy=x^{-1}y^{-1}=(yx)^{-1}=yx.$$

answered Dec 27 '16 at 17:47

Bumblebee

2 Answers2