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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] $ have involutions?

I think doesn't have!

Help!

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    It means that $\phi$ stabilizes $\mathbb{Z}$, so for $(a,b) \in V\times \mathbb{Z}$, $(a,b)^\phi=(a^\phi,b)$.2012-08-23

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