# Universal property of $N\rtimes K$

Given groups $N$ and $K$, if $K$ acts on $N$ by $$K\xrightarrow{\theta}\operatorname{Aut_{Grp}}(N),$$ then we can define a group $N\rtimes_{\theta}K$ whose elements are like in $N\times K$ but the multiplication is defined by $$(n,k)(n',k')=(n\theta_{k}(n'),kk')$$ 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 $$1\longrightarrow N\longrightarrow N\rtimes_{\theta} K\longrightarrow K\longrightarrow 1.$$ 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?

Thanks!

-
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 at 16:26