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The title pretty much says it all... anyway... let $G$ be a semi-direct product of $N$ by $Q$, and let $H$ be a subgroup of $G$.

Can one always find subgroups $N_1$ and $Q_1$, of $N$ and $Q$ respectively, such that $H$ is the semi-direct product of these two groups?

If not in general, can one say anything about the following cases:
i) $G = \text{GL}(n,K)$, i.e. the general linear group of dimension $n$ over some field $K$
ii) $G = \text{Aff}(E)$, i.e. the group of all affine motions of some (finite dimensional) linear space

thx in advance

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    For case i) do you really want $G = \mathrm{GL}(n, K)$ or did you intend to write $Q = \mathrm{GL}(n, K)$?2011-06-04
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    Thanks for your input. I'm embarresed by the triviallity of the counterexamples. @Jug: No I don't, why (honestly I don't understand the question, I think...) Also: About the infinite cyclyc subgroups of Aff(E)...it supersedes my imagination, when I think of the homgenized version of Aff(E), i.e. the embedding into GL(n+1,E), how such a subgroup, i.e. one that is not of the form mentioned in my first post, can arise. Therefor I'd really appreciate it, if Jim Belk, or someone else, could elaborate on that. Lastly: I was thinking about GL(n,E) being the semi-direct product of SL(E) by K*.2011-06-06
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    (The previous comment was converted from a "comment as answer" posted by the OP, and was edited slightly for length. 10K users can see the original below.)2011-09-05

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