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Questions tagged [semidirect-product]

The semidirect product is a construction in group theory generalizing the direct product. It arises as the structure of a group $G$ with a normal subgroup $N$ having a complement $N$.

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Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
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Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set $X$...
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$G\rtimes_\phi H \cong G\rtimes_\psi H$ when certain automorphisms exist

I'm reviewing for exams and came across this problem from an older exam: Let $G$ and $H$ be groups and let $\phi,\psi : H\to \operatorname{Aut}(G)$ be homomorphisms from $H$ into the group of ...
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Relation between kernels of homomorphisms and the semidirect product of groups

I'm reading Abstract Algebra of Dummit and I got stuck on this point. When we try to build the semi-direct product of 2 group predefined, such as $H$ and $K$, then all the semi-direct product groups ...
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Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, \mathbb{Z}/p\mathbb{Z})$

This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous ...
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Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$

Let $n = p_1 p_2 \cdots p_k$ be the product of pairwise distinct odd primes. Let $X$ be the set of element in $\operatorname{Aut}(\mathbb Z/n\mathbb Z)$ of order $1$ or $2$. For each $\psi\in X$, the ...