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Let $F_\infty$ be the free group on infinitely many generators, and let $\phi: Z \rightarrow$ Aut($F_\infty$) be any group homomorphism. If we form the semi-direct product $F_\infty \rtimes_\phi Z$, could this group be isomorphic to $F_\infty$?

Thanks! Kevin

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    Also asked and answered [on MO](http://mathoverflow.net/questions/106472/could-f-infty-rtimes-z-be-isomorphic-to-f-infty). If you ask the same question on several sites, it would be nice to provide links between the various versions of the question to avoid duplication of effort.2012-09-06

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Um how about take $Z = \{e\}$?

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    The $Z$ is most likely intended to be the infinite cyclic group $\mathbb{Z}$, as shown by [the MO version](http://mathoverflow.net/questions/106472/) of the same question.2012-09-06