# Subgroup of the internal semidirect product [closed]

Let $$H$$ and $$K$$ be a finite groups and $$G'$$ be a normal subgroup of the internal semidirect product $$H.\,K$$. Take $$\,H'=\,H \cap \,G'$$ and $$K'= (G' \,H) \cap \,K$$. We can see easely that $$\mid G' \mid =\mid H'\mid \mid K'\mid$$.

1. Is it true that $$G'=H'.\,K'$$.
2. In the case that the answer is "no": Are there any reasonable constraints to $$H$$ and $$K$$ such that the answer is "yes"?

Any help would be appreciated so much. Thank you all.

## closed as off-topic by Chris Godsil, YCor, Derek Holt, Gabriel C. Drummond-Cole, Pace NielsenFeb 11 at 17:38

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• In general $K'$ is not a subgroup of $G'$. For example $G'$ could be a complement of $H$ that is disjoint from $K$. That can happen even with a direct product. – Derek Holt Feb 11 at 10:26