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 $\mid G' \mid =\mid H'\mid \mid K'\mid$.
- Is it true that $G'=H'.\,K'$.
- 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.