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In my group theory course, I came across the following types of questions:

(1) Show that $N\rtimes_{\phi} H$ is direct if and only if $\phi$ is trivial if and only if ${e_N}\times H\triangleleft N\rtimes_{\phi} H$.

or

(2) Write $\text{GL}_n(k)$ as a natural semi-direct product. When is it direct?

And this always confuses me a bit. What do we actually mean by asking "is this semi-direct product direct"?

For example in (1), do we need to see when there is an abstract isomorphism $N\rtimes_{\phi} H\cong N\times H$, or is it the question in which cases the natural application $(n,h)\mapsto (n,h)$ is an isomorphism?

And in the second question, we see that $\text{GL}_n(k)\cong \text{SL}_n(k)\rtimes k^\times$ if we identify $k^\times$ with a suitable subgroup of $\text{GL}_n(k)$, e.g. $\{\text{diag}(\alpha,1,...,1)\mid \alpha\in k^\times\}$ or if $\lambda\mapsto \lambda^n$ is an automorphism of $k^\times$, then we can also take $\{\alpha I_n\mid \alpha\in k^\times\}$.

The difference between (1) and (2) seems to be, that in (1) we already fixed the underlying automorphism, and in (2) we didn't, so the question becomes wether there is one for which the product is direct, in whatever sense.

So: is there a general convention for what this meens? Or can it depend from the context of the question?

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  • $\begingroup$ The first question is definitely supposed to mean in which cases the natural application is an isomorphism. Indeed, it is not too hard to come up with counterexamples otherwise, i.e., with groups that can be written as $A\rtimes_\phi B$ with non-trivial $\phi$ as well as $A\times B$ $\endgroup$ – Hagen von Eitzen Nov 1 '17 at 11:36
  • $\begingroup$ I agree that it is a confusingly worded question. In (2), the semidirect product decomposition that you have given is never direct when $n>1$. But there are cases in which ${\rm GL}_n(k)$ is a nontrivial product, $\endgroup$ – Derek Holt Nov 1 '17 at 13:26

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