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Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate student was asking me about this, having dug a version out of an obscure research paper in the process of studying an unrelated technical problem. I'm not sure what exists in group theory books or other such sources. The answer seems to have the flavor of cocycles and coboundaries but is apparently somewhat complicated to write down.

Is there a convenient reference giving a recipe for all subgroups of a finite group written as a semidirect product of two known groups relative to a known action of one on the other?

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    $\begingroup$ Thevenaz has a (rather elementary) paper discussing maximal subgroups of direct products (J. Alg, 1997). He points out that subgroups of GxH can be described in terms of graphs of homomorphisms G->H. There might be some sort of generalization to semi-direct products using some kind of twisted homomorphisms? I recall having thought about this years ago, but I don't think I got anywhere. $\endgroup$
    – Dan Ramras
    Commented Jul 9, 2012 at 23:28
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    $\begingroup$ I'll also point out that the mathreviews entry for the Thevenaz paper ("Maximal subgroups of direct products") says that the description of subgroups of G x H was earlier in Suzuki's "Group Theory I". On the other hand, the Thevenaz paper is now freely available from any internet connection, while not everyone has Suzuki's book. $\endgroup$
    – Russ Woodroofe
    Commented Jul 9, 2012 at 23:38
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    $\begingroup$ An idle web search found that the same question came up on math.stackexchange.com a couple of years ago. See math.stackexchange.com/questions/5153/… . In addition to the Usenko result that Bugs gives below, a reference is also given there to a paper of Rosenbaum "Die Untergruppen von halbdirekten Produkten". $\endgroup$
    – Russ Woodroofe
    Commented Jul 10, 2012 at 19:23
  • $\begingroup$ @Russ: It didn't even occur to me to check stackexchange after looking over past entries here. I'm not sure what the dividing line between the two sites is, but the question did strike me as somewhat sophisticated while also probably "standard" in the literature. I can access a very short review of Rosenbaum (but not the article). $\endgroup$
    – Jim Humphreys
    Commented Jul 10, 2012 at 21:52
  • $\begingroup$ Suzuki's "Group Theory" volume 1, Chapter 2, Section 7 is devoted to "Extensions of groups and cohomology theory". It's got a lot of good stuff in it (38 pages worth) but I don't know if it's got what your student needs... Worth a browse though if s/he hasn't looked at it already. $\endgroup$
    – Nick Gill
    Commented Jul 18, 2012 at 9:39

1 Answer 1

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Usenko, Subgroups of semidirect products, Ukrainian Mathematical Journal, 1991, Volume 43, Numbers 7-8, Pages 982-988

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    $\begingroup$ The translation is accessible online through our library, so I can see that it contains a kind of answer. I was hoping for something more readable, say in a group theory book, but that was probably naive to expect. $\endgroup$
    – Jim Humphreys
    Commented Jul 10, 2012 at 21:54

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