Suppose we have a semidirect product $G = N \rtimes K$, (here $N$ is the normal subgroup). I know that in general, questions about subgroups of $G$ are hard to answer, and that we don't have any nice theorem like Goursat's lemma for direct products.

However I was wondering: what happens if we ask some more specific questions? In particular, I would like to know: what are the subgroups of $G$ which are isomorphic to $K$ and intersect $N$ trivially? Are there any besides the conjugates of $K$? An answer with any single one of those two conditions is very welcome as well. By the way, I am working with finite groups.

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    $\begingroup$ Yes there are certainly semidirect products in which not all complements of $N$ are conjugate. There are small easy examples when $G$ is a direct product. When $N$ is abelian, the number of conjugacy classes of complements is equal to the order of the first cohomology group $H^1(G,N)$. $\endgroup$ – Derek Holt Oct 14 '17 at 8:11

Your question is equivalent to say that what are all complements of $N$ in $G$?

Answer of this question is still difficult. But there is a theorem when $(|K|,|N|)=1$.

In that case, it is known that if $H$ is also a complement of $N$ then $H=K^n$ for $n\in N$. This theorem is called as Schur-Zassenhaus theorem.


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