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It's not hard to imagine early group theorists getting the inspiration for the semidirect product because after you've seen a few examples of finite nonabelian groups, the pattern starts to emerge on its own.

But who first codified the definition, explicitly proposed looking at a mapping $\varphi :H\to Aut(N)$, and showed that $\langle n, h\rangle\langle n', h'\rangle = \langle n\varphi_{h}(n'), hh'\rangle$ gives a group operation on the product set $N\times H$, and when?

I'd be interested in any leads on any part of this: earlier prefigurings and special cases; later distillations; who coined the name; etc.

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    $\begingroup$ And to be clear: You're only interested in the "homomorphism into automorphism group" point of view? Because the Schur-Zassenhaus theorem on semidirect products was proven by Zassenhaus in 1937. However, I'm willing to bet quite a lot that Burnside used something equivalent to semidirect products (in his 1897 book, something about whether two groups are "permutable" is said, a bit after he defines direct products. I can't tell, but I think he might be giving an old definition for a semidirect product). $\endgroup$ – pjs36 Dec 22 '16 at 22:25
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    $\begingroup$ To be clear, I'm not sure what I'm interested in because the history is so murky and I've had so little success finding anything at all. I'd like as much material as possible. $\endgroup$ – MJD Dec 22 '16 at 22:28
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    $\begingroup$ In my experience, that is not the case. $\endgroup$ – MJD Mar 24 '17 at 14:10
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    $\begingroup$ Progress report: in pursuing this question I have been slowly working my way through Burnside's 1897 Theory of Groups of Finite Order. If the semidirect product was known then, one would expect to find it there. It is hard to be sure, because Burnside's terminology is so different from modern terminology, but so far I have found nothing like a semidirect product, even in the places I would expect to. For example, Burnside constructs the holomorph of a group, but the construction is very 19th-century and does not appear to involve a semidirect product. $\endgroup$ – MJD Mar 24 '17 at 14:18
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    $\begingroup$ ... The group operation on the union is simply composition of permutations, and the holomoprh is the closure of the union with respect to this operation. No doubt there is a way to view this as a semidirect product but it seems clear to me that Burnside didn't view it that way. He certainly doesn't mention it as being a special case of any sort of semidirect product construction. $\endgroup$ – MJD Mar 28 '17 at 18:57
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This question, who invented the semidirect product (and, the holomorph for that matter) , interested me for a long time, too. Two people might be good candidates: G.A. Miller and O. Hoelder. Miller wrote papers on the holomorphs of cyclic groups early in the 1900's and he was a contemporary of Hoelder (a pioneer in considering automorphisms).

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  • $\begingroup$ Sorry about the earlier comment. I misread the question. Some others misread something else. Welcome to Math.SE! $\endgroup$ – Jyrki Lahtonen Sep 18 '17 at 17:29

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