Goursat's Lemma characterizes the subgroups of direct products. Is there a similar characterization for the subgroups of semidirect products? What about if I'm only interested in the normal subgroups?
What are the subgroups of a semidirect product?
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1If you have a non-split semi-direct product, then there are not two projections, only one. So any such characterization can't be *too* similar. – 2010-09-21
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2@MattE - I assume this means a semidirect product $K\rtimes Q$ with nontrivial action of $Q$ on $K$ (i.e. that is not a direct product). All semidirect products are split, no? – 2016-10-09
2 Answers
The short answer is that there is nothing nearly so nice as Goursat's Lemma. You can certainly reduce easily to the case where $\pi_K(H)=K$, much like you can reduce Goursat's Lemma to the case of a subdirect product, but after that it gets complicated. To give you an idea, here are three references.
A theorem of Rosenbaum (Die Untergruppen von halbdirekten Produkten, Rostock. Math. Kolloq. No. 35 (1988), 21-30) gives (from the MathScieNet Review MR991728 (90c:20032)):
Theorem. A set $U$ of elements of the semidirect product $G=NK$ with $N\triangleleft G$ is a subgroup of $G$ if and only if
- $UN\cap K$ and $U\cap K$ are subgroups of $G$;
- $U\cap N$ is a subgroup and $UK\cap N$ is a collection of $U\cap N$-cosets in $N$; and
- There is a mapping $\varphi$ defined for all $g\in UK\cap N$ mapping $(U\cap K)g$ onto some coset $n(U\cap N)$, with $n\in N$, satisfying $\varphi(g_1g_2)=g_2^{-1}\varphi(g_1)g_2\varphi(g_2)$.
The criterion was then used by Gutiérrez-Barrios to develop a criterion for a set of elements to be a normal subgroup of the semidirect product (Die Normalteiler von halbdirekten Produkten. Wiss. Z. Pädagog. Hochsch. Erfurt/Mühlhausen Math.-Natur. Reihe 25 (1989), no. 2, 108-114. MR1044548 (91b:20029))
Usenko (Subgroups of semidirect products, english translation in Ukrainian Math. J. 43 (1991), no. 7-8, 982-988 (1992), MR1148867 (92k:20045)) uses crossed homomorphisms to study subgroups of semidirect products.
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0Just a question. $UK \cap N$ is a set of elements, so how could this be a collection of cosets (which would be like a set of sets), am I missing something? And out of curiousity, wouldn't it be interesting to look at $U/U\cap N \cong UN/N$ which would be isomorphic to a subgroup of $K$ (if $U$ forms a subgroup), or is this just a trivial observation... – 2018-06-28
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0@StefanH: It is not described as a collection of cosets. For each *element* $g\in UK\cap N$, you have a coset $(U\cap K)g$ of $U\cap K$. The map $\varphi$ sends this coset to a coset of $U\cap N$. – 2018-06-28
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0Okay, thanks for clarification. – 2018-06-29
While searching on the net i found this article. See whether it helps or not: http://www.springerlink.com/content/l272110h87u05667/fulltext.pdf
If one can't access this then try: dx.doi.org/10.1007/BF01058705 or ams.org/mathscinet-getitem?mr=1148867
Thanks to Jack.
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2The reference is: **Usenko**, V. M. "Subgroups of semidirect products." Ukrain. Mat. Zh. 43 (1991) 1048–1055. (translated as) http://dx.doi.org/10.1007/BF01058705 http://www.ams.org/mathscinet-getitem?mr=1148867 – 2010-09-21
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0@Jack Schmidt: Yes Jack whats the problem with that – 2010-09-21
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4@Chandru: not everyone has access to the springer full text, and it's always nice to give the reference in a readable format. Jack is providing an English translation and a link to the MathReview. He is *complementing* your reply, not criticising it. – 2010-09-21
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0@Arturo Magidin: I am not saying anything against him Arturo, i actually didnt know what was the problem. – 2010-09-21