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YCor
YCor
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finite Perfect group that is split extension of a normal free subgroup of non perfect groupfinite index

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semi directsemidirect product $F\rtimes L$ is perfect? Thanks

Thanks @YCor for reformulating the question.

finite subgroup of non perfect group

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semi direct product $F\rtimes L$ is perfect? Thanks @YCor for reformulating the question.

Perfect group that is split extension of a normal free subgroup of finite index

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semidirect product $F\rtimes L$ is perfect?

Thanks @YCor for reformulating the question.

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tota
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Let $A$ beDoes there exists a non-perfecttrivial free group such that it is semidirect product of $G$$F$ and $H$ where $G$ is a finite group and $H$ is a free group of finite rank. My question is can$L$ acting on $G$ be a perfect group? If not can we prove$F$ such that? Any comments suggestions regarding this question the semi direct product $F\rtimes L$ is highly appreciated.perfect? Thanks in advance@YCor for reformulating the question.

Let $A$ be a non-perfect group such that it is semidirect product of $G$ and $H$ where $G$ is a finite group and $H$ is a free group of finite rank. My question is can $G$ be a perfect group? If not can we prove that? Any comments suggestions regarding this question is highly appreciated. Thanks in advance.

Does there exists a non-trivial free group $F$ and a finite group $L$ acting on $F$ such that the semi direct product $F\rtimes L$ is perfect? Thanks @YCor for reformulating the question.

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tota
tota
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finite subgroup of non perfect group

Let $A$ be a non-perfect group such that it is semidirect product of $G$ and $H$ where $G$ is a finite group and $H$ is a free group of finite rank. My question is can $G$ be a perfect group? If not can we prove that? Any comments suggestions regarding this question is highly appreciated. Thanks in advance.