# What are the subgroups of a semidirect product?

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?

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If 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. – Matt E Sep 21 '10 at 15:25

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

1. $UN\cap K$ and $U\cap K$ are subgroups of $G$;
2. $U\cap N$ is a subgroup and $UK\cap N$ is a collection of $U\cap N$-cosets in $N$; and
3. 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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