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There is a definition which help us to understand the semi-direct product well:

Let $K$ be a subgroup of a group $G$. Then a subgroup $Q\leq G$ is a complement of $K$ in $G$ if $K\cap Q=1$ and $KQ=G$.

Habitually, whenever I see the form $KQ=G$, I think of one of the subgroups $K$ or $Q$ are at least normal in group. But in the definition above J.J.Rotman quoted that $K$ is not necessarily normal in $G$ and moreover he didn't say anything about normality for $Q$ as well.

May I ask you to present me an example, having two subgroups not normal in $G$, and yet $G=KQ$? And if I misunderstand about above, please tell me. Thanks.

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2 Answers 2

up vote 3 down vote accepted

Take $G = A_{5}$,$K = A_{4}$ (say the subgroup of $A_{5}$ fixing $5$) and $Q$ to be a Sylow $5$-subgroup of $A_{5}.$

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There are many examples: When $p$ is an odd prime, you can take $G=S_{p},$ $K = S_{p-1}$ and $Q$ the subgroup generated by a $p$-cycle. On the other hand, such factorizations are in a sense fairly unusual in simple groups,(see work of J. Saxl for example). –  Geoff Robinson Sep 16 '12 at 7:34
Thanks for the example. Now, I can see the solution. Sorry for asking but; is there a difference between being a complement of a subgroup and being complementary to each other. See Thanks for the time. –  Babak S. Sep 16 '12 at 7:44
As for the second question, it is a matter of language and definition. My interpretation would be that I would expect them to mean the same thing, but I have not looked at the link –  Geoff Robinson Sep 16 '12 at 7:50

This is sometimes called a(n internal) Zappa-Szép product. There are examples at the Wikipedia article.

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