# Some questions on semi-direct product

Question 1

Let $G$ be a group and $H\unlhd G$ and $K\leq G$ such that $H\cap K = 1$ and $HK=G$

Let $\varphi:K\rightarrow Aut(H)$ be a homomorphism such that $HK\cong H\rtimes_\varphi K$.

Then is $\varphi$ trivial if $K\unlhd G$? And is $K\unlhd G$ if $\varphi$ is trivial?

Question 2

Let $H,K$ be groups and $\varphi_1,\varphi_2:K\rightarrow Aut(H)$ be homomorphisms such that $H\rtimes_{\varphi_1} K\cong H\rtimes_{\varphi_2} K$.

Then, is there a relation between $\varphi_1$ and $\varphi_2$?

In general, if $\phi=\varphi\pmod{\operatorname{Inn}(H)}$ then $H\rtimes_{\phi}K\cong H\rtimes_{\varphi}K$, where $\operatorname{Inn}(H)$ denotes the inner automorphisms of $H$ (automorphisms of the form $\gamma_g: h\mapsto g^{-1}hg$).
The answer to your second question is more subtle. Under certain circumstances (for example, $H$ does not map onto $\mathbb Z$ and $K$ is infinite cyclic) the above conditions are necessary and sufficient. For a counter-example, look up the classification of groups of order $pq$ ($p, q$ prime. There I no more than two groups of order $pq$, but there are more than two relevant maps (and no non-trivial inner ones!).
• And I think I should edit the hypothesis as $K\unlhd HK$ rather than $K\unlhd G$ – Rubertos Feb 18 '15 at 8:20