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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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142 views

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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290 views

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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Why is $\mathbb{Z_6}=\mathbb{Z_3} \rtimes \mathbb{Z_2}$?

Semidirect product must satisfy 3 axioms: (1) $N \unlhd G$ (2) $N\cap H = 1$ (3) $NH = G$ But clearly $\mathbb{Z_3} \cap \mathbb{Z_2} = \{0,1\} \neq \{0\}$, and it seems to be the case that $\mathbb{...
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112 views

$U(n)=SU(n)\rtimes U(1)$?

Wiki says that the group $U(n)$ is a semi-direct product of $SU(n)$ and $U(1)$. Each element $g$ of a semi-direct product $G=HK$, should be uniquely represented as $g=hk$. $SU(n)$ and $U(1)$ have ...
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354 views

The holomorph of $Z_2 \times Z_2$

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184, Exercise 5): Let $G=\text{Hol}(Z_2 \times Z_2)$ (a) Prove that $G=H \rtimes K$ where $H=Z_2 \...
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Semidirect product, normal subgroup exercise

Let $G$ be a group and let $H,K$ be subgroups of $G$ such that $G=H \rtimes K$. (i) Show that if $K \lhd G$, then $kh=hk$ for all $h \in H, k \in K$. (ii) Deduce that $G$ is abelian if and only if $...
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semidirect product problem

I am trying to show that the semidirect product of $G$ with $G$, where $G$ is a finite group, with the automorphism by conjugation on itself, is isomorphic to direct product of $G$ with $G$. Please ...
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If $q$ is not a divisor of $p-1$, then $G$ is cyclic. An open discussion on different proofs.

Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If $q$ is not a divisor of $p-1$, then $G$ is cyclic. I know the proof using Sylow's theorem but I was wondering if it can be proved ...
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Classifying all non-abelian groups of order $55$

Suppose a group $G$ is the semidirect product of normal subgroup $N$ and subgroup $H$, i.e., $G=N\rtimes_\varphi H$. Find all semidirect products (up to isomorphism) of $N=\mathbb Z_{11}, H=\mathbb ...
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170 views

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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286 views

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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119 views

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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488 views

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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55 views

Isomorphism factor by factor

Consider groups are finite. Let $G_1 = A \rtimes_{\phi_1} B_1$ and $G_2 = A_2 \rtimes_{\phi_2} B_2$. Note that $A_1,A_2,B_1,B_2$ are cyclic groups. It is also known that $A_1 \cong A_2$ and $B_1 \...
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36 views

Semidirect product of subgroups

Let $G$ be a group with $N \unlhd G$ such that there is $K \leq G$ with $KN = G$ and $K \cap N = 1$. It is well known that in this case $$G \cong N \rtimes_\phi K$$ where $\phi: K \to Aut(N): k \...
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62 views

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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219 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi H \;...
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73 views

General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
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636 views

Symmetric group isomorphic to semidirect product of Alternating group and Z/2Z

I'm having a hard time understanding why $A_n \rtimes \mathbb{Z}_2 \cong S_n$. I understand that $A_n$ is normal in $S_n$. But that's about it. What would the $\alpha$: $\mathbb{Z}$$_2$$\...
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138 views

How can you tell if a normal subgroup induces a semidirect product?

Suppose I have some (finite) group $G$ and a normal subgroup $N$. I know there's no full characterization of whether $G \cong N \rtimes G/N$, but are there well-known tests I can use to answer the ...
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121 views

How does action by conjugation determine the product stucture of a semidirect product?

Consider $G=P\ltimes Q$ where $P\cap Q=\{e\}$ and $Q<N_G(P)$. Here $\ltimes$ is the inner semidirect product. Here I believe it is the case that the conjugation action of $P$ on $Q$ will determine ...
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$AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = ...
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Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
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Does the interior semidirect product of Lie groups $G = N \rtimes H$ respect the projection $G \to H$?

Suppose we have some matrix Lie groups $N$ and $H$ both subsets of the $n \times n$ matrices, and a matrix Lie group $G$ for which we know $G = N H$, $N$ is a normal subgroup of $G$, and $N \cap H = \...
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Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
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How to prove that semidirect product of $Z_{13}$ and $Z_3$ is non Abelian for a non-trivial homomorphism

The semidirect product of $Z_{13}$ and $Z_3$ is given here Finding presentation of group of order 39 as $\{x,y | x^{13} = y^3 = 1, yxy^{-1} = x^3\}$. I understand how this is arrived at but to show ...
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Groups in the generalized triple “semidirect” product of semigroups

A semigroup $S$ acts on another semigroup $V$ (written additively for better readability, but could be non-commutative) on the left if $$ s(v_1 + v_2) = sv_1 + s v_2, \quad s(s')v = (ss')v $$ for $s,...
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Constructing a semidirect product in MAGMA

I am trying to construct in MAGMA the group $$ (C_3 \times C_3) \rtimes Q_8$$ in which the action of $Q_8$ on $C_3 \times C_3$ is given by the center $Z(Q_8)$ acting trivially, and the quotient $Q_8 /...
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1answer
121 views

Existance of a normal $p$-complement. (5C.12 Finite Group Theory, Isaacs)

Let $G$ be a finite group, $N$ normal subgroup with index in $G$ divisible by $p$ prime and suppose that a Sylow $p$-subgroup of $G$ is cyclic. Then $N$ has a normal $p$-complement. This is the ...
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337 views

Direct Product and Semi-direct Product between $S_{3}$ and $Z_{2}$, and Related Problem

I am trying to figure out the following question: 0) Is the direct product $S_{3}\times Z_{2}$ isomorphic or non-isomorphic to the semi-direct product $S_{3}\rtimes_{\phi}Z_{2}$ where $\phi:Z_{2}\...
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167 views

Find a non-trivial outer automorphism of $A_n$?

The symmetric group $S_n$ can be decomposed as a semi-direct product of the alternating group $A_n$ and a subgroup of $S_n$ of order 2. Use this fact to find a non-trivial outer automorphism of $...
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1answer
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About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$ [closed]

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? ...
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1answer
50 views

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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1answer
407 views

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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2answers
248 views

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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39 views

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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1answer
70 views

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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1answer
231 views

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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1answer
42 views

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 ...
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1answer
49 views

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 ...
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1answer
91 views

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 \...