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Given groups $N$ and $K$, if $K$ acts on $N$ by \begin{equation} K\xrightarrow{\theta}\operatorname{Aut_{Grp}}(N), \end{equation} then we can define a group $N\rtimes_{\theta}K$ whose elements are like in $N\times K$ but the multiplication is defined by \begin{equation} (n,k)(n',k')=(n\theta_{k}(n'),kk') \end{equation} where $\theta_k\in \operatorname{Aut_{Grp}}(N)$ is the image of $k$ under $\theta$.

This semidirect product is very useful in studying structures of finite groups, especially it solves the extension problem \begin{equation} 1\longrightarrow N\longrightarrow N\rtimes_{\theta} K\longrightarrow K\longrightarrow 1. \end{equation} But I am wondering whether this can be defined using universal properties? In the abelian case $N\rtimes K$ is just $N\times K$ so we have the universal property of products, but what about the nonabelian case?


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I'd guess that $N\rtimes K$ is universal omong all $N\to G\leftarrow K$ where (the image of) $K$ operates on (the image of) $N$ via $\theta$ –  Hagen von Eitzen Jan 22 '13 at 16:26
mathoverflow.net/questions/96078/… –  user26857 Jan 22 '13 at 16:31
@MartinBrandenburg Please consider converting your MO answer to an M.SE answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. –  Julian Kuelshammer Jun 23 '13 at 8:01
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