# Involutions in Group

Set $\varphi \in \operatorname{Aut}\left( G\right)$ with order $2$. Consider the semidirect product $G\rtimes \left\langle \varphi \right\rangle$. ($G$ is infinite)

Let $\left[ G,\varphi \right] =\left\langle g^{-1}g^\varphi ;g\in G\right\rangle \trianglelefteq G.$

My question is: Does $\left[ G,\varphi \right]$ has involutions?

I think doesn't have!

Help!

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What?? Let $G$ be the Klein four group, then $\left[ G,\varphi \right]$ is a group of order 2... –  user641 Aug 22 '12 at 22:09
I've voted to close. Your original question was easily seen to be false, and after I gave a counterexample, you changed the question. You changed it without thinking it would seem, because my counterexample can easily be extended to cover this new case as well. I am not confident you have given much thought to this question, and without any motivation, I voted to close. –  user641 Aug 22 '12 at 22:17
your answer for finite groups was good. Although it didn't place before, my subject is really for G infinite.Sorry! –  User2040 Aug 22 '12 at 22:25
Just to follow up on Steve D's comment, you can cross the Klein four group with $\mathbb{Z}$ and just let $\phi$ be the identity on the $\mathbb{Z}$ part. Before you edit again, consider taking $G$ to be an infinite direct sum of copies of $\mathbb{Z}_2$. –  MartianInvader Aug 22 '12 at 22:26
What does mean $\varphi$ let be identity on the Z part? –  User2040 Aug 22 '12 at 22:58