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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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“Super-complement” of a normal subgroup

$\DeclareMathOperator{\Aut}{Aut}$Consider a group $G$ with a normal subgroup $N\triangleleft G$ and suppose that $G$ has a subgroup $H$ so that $G = HN$. Then $H$ acts on $N$; in other words, we have ...
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Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
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Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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Isomorphic semidirect products

Suppose that $\rho : K \to K$ is a group automorphism of $K$, and $\psi : K \to \operatorname{Aut}(H)$. Show that $H \rtimes_\psi K \cong H \rtimes_{\phi} K$ where $\phi = \psi \circ \rho$. Just a ...
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Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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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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Find the semidirect products of $C_2\times C_2$ by $C_3$

Find the semidirect products of $C_2 \times C_2$ by $C_3$, that is: $(C_2 \times C_2) \rtimes C_3$ My approach: I let $C_3:=\langle y\rangle$ and let $\phi : C_3 \to \mathrm{Aut}(C_2 \times C_2)$ be ...
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In what case the semi direct product is abelian?

Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian?
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When does a $p$-group split as semidirect product with its commutator?

Given a (finite) group $G$ and its commutator subgroup $G'$, when is it the case that $$1\to G'\to G\to G/G'\to 1$$ splits? Specifically, can we say anything if we add the assumption that $G'$ is ...
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I would like to construct a group that I have written down on paper in GAP.

First, let $V=\text{GF}(2^{11})$ (the group under addition) and let $\sigma$ be the squaring map (Frobenius map). Since $p=23$ divides $2^{11}-1$ there exist a $p^{\text{th}}$ root of unity in $V$, ...
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Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...
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Question about classifying semidirect product

I have in some notes, this statement: Given $C_3\ltimes C_7$ we know that for $a\in C_3$ and $b\in C_7$, and some $k$: $$aba^{-1}=b^k$$ $$k^3\equiv 1(7)$$ The reason given is that $a^3=1$. ...
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About the construction of semidirect products

I need help with the following question: We have $G_0=H_0[N]_\rho$ a semidirect product, and $\pi:H\to H_0$ an epimorphism such that $K=\ker(\pi)\unlhd H$, and $H/K\cong H_0$. We have to construct a ...
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Exercise about semidirect product

This is exercise 7.12 from Algebra, Isaacs. $G= N \rtimes H \$ is a semidirect product; no nonidentity element of H fixes any nonidentity element of N; identify N and H with the corresponding ...
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GAP semidirect product

I am newbie in the GAP and in the group theory. Now I am trying to make semidirect product if GL(3,2) and GL(3,2) inversed and transposed. I use code below ...
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A short but tricky question on Sylow theorem and semidirect product

Let $G:=H \rtimes_{\phi} K$, $\phi: K \to \mathrm{Aut}(H)$ and is homomorphic. There is only one $p$-Sylow subgroup of $G$ (denoted by $P$). $p$ divides $|\phi(K)|$. Prove $p$ divides $|H|$. *This ...
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What does it mean for a subgroup to fix a vector space?

I was wondering if anyone could explain what is meant by the two statements below? Any help would be greatly appreciated. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of ...
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Classifying groups of order $2p^2$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 185 Exercise 15): Let $p$ be an odd prime. Prove that every element of order $2$ in $GL_2(\mathbb{F}_p)$ ...
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Frobenius group as semidirect product of finite group with a regular group of automorphisms.

Let $G$ be a finite group. We say a non-trivial group of automorphism $A$ on $G$ is regular, if each non-trivial automorphism of $A$ is regular, i.e. fixes only the identity. It is remarked in ...
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Solvable groups as iterated semidirect products of supersolvable groups

All my groups are assumed to be finite. My question is the following: Is every solvable group an iterated semidirect product of supersolvable groups? A group $G$ is said to be supersolvable if ...
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On subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$ [closed]

In order to study the subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$, I need to solve the following exercice from the Book (Dummit & Foote p101): Exercice: Let $H$ be a ...
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Automorphisms of a semidirect product.

Is there any way to compute $Aut(H \rtimes K)$? I don't know either how to compute the simplest ones, like semidirect product of two cyclic groups. Any hint, books to read or articles about this theme?...
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Conjugates of semidirect product complements

Suppose we have a finite group $G$ which contains a normal subgroup $M$. Consider the following subgroups of $G$: $C, Q_1, \ldots, Q_{|M|}$, all of which are different complements of $M$ in $G$ (that ...
Semidirect products of $V_4 \ltimes_\alpha C_3$ and isomorphies
My task is to find all semidirect products $V_4 \ltimes_\alpha C_3$ and to find those, who are isomorphic. First of all, I've got to find the automorphism group of $C_3$. I know, that it is ...
Semidirect product $\mathbb{F}_q \rtimes \mathbb{F}_q'$
Let $\mathbb{F}_q$ be an additive group of finite field and $\mathbb{F}_q' \simeq \mathbb{Z}_{q-1}$ be a multiplicative group of finite field. I want to build a semidirect product \$\mathbb{F}_q \...