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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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Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
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What is the center of a semidirect product?

Let $G_1$ and $G_2$ be groups. Let $\varphi:G_2\rightarrow \operatorname{Aut}(G_1) $ be a group homomorphism defining the semidirect product $G_1 \rtimes G_2$. Determine the center $\operatorname{Z}(...
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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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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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$\mathbb S_n$ as semidirect product

In this note, I've read that $\mathbb S_n$ is a semidirect product of the alternating group $A_n$ by $\mathbb Z_2$. So I am trying to define a morphism $\rho: \mathbb Z_2 \to Aut(A_n)$ to show that $\...
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Semidirect Products with GAP

I'm wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I'm trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. ...
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When does a group of dilations/scalings exist in a metric space?

Notation: Let $(X,d)$ be a metric space. A similitude will be (by convention) a surjective (hence bijective) map $f: X \to X$ such that for all $x_1, x_2 \in X$, $d(f(x_1),f(x_2)) = r d(x_1, x_2)$ for ...
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Semi-direct v.s. Direct products

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct ...
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GAP semidirect product algorithm

Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group? EXAMPLE: ...
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Semi-direct product Lie algebra

Given Lie algebras $S$ and $I$ and a Lie homomorphism $\theta:S\to Der I$, we have the semidirect product to be the space $S\oplus I$ with operation $$ (s_{1},x_{1})(s_{2}x_{2}):=([s_{1},s_{2}],[x_{1},...
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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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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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What is the motivation for semidirect products?

I haven't the slightest idea why (inner or outer) semi-direct group products are an interesting construction. I understand inner direct products, because you're just giving conditions for which a ...
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semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function $f:S\to\...
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Classify groups of order $pq^2$ using semidirect product

I am struggling with semidirect products and how they can be used to classify groups of a certain order. In particular, I need help with the nonabelian case. This is the problem I am working with.. ...
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Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$?

Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$? I can see that $\varphi(K) = \varphi(\phi(K))$, but it is not clear to me how the ...
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Presentations of Semidirect Product of Groups

I have seen here that given two groups $G=\langle X|R \rangle := F(X)/N(R)$ and $H=\langle Y|S \rangle := F(Y)/N(S)$, then their semidirect product can be written as: $$ G\rtimes_\phi H \;=\; \langle ...
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Nonabelian groups of order $p^3$

From a little reading, I know that for $p$ and odd prime, there are two nonabelian groups of order $p^3$, namely the semidirect product of $\mathbb{Z}/(p)\times\mathbb{Z}/(p)$ and $\mathbb{Z}/(p)$, ...
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Is semidirect product unique?

This is about semi direct product on Dummit and Foote algebra text book. Why is this statement true? Theorem 12. Suppose $G$ is a group with subgroups $H$ and $K$ such that $H\...
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$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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1answer
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Classifying groups of order 60

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 185, Exercise 14): This exercise classifies the groups of order $60$ (there are thirteen isomorphism types)....
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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 ...
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Exercise 3A.7 of “Finite group theory”, M. Isaacs

Let $G$ finite group and $\sigma \in \text{Aut}(G)$, suppose that at most two prime numbers divide $o(\sigma)$. Show that $\left \langle \sigma \right \rangle$ has a regular orbit on $G$. Suppose $o(\...
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Subgroups of the Semi-Direct Product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$

I want to list all the subgroups of the semi-direct product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$, under the homomorphism $\theta: (\mathbb{Z}/7\mathbb{Z})^{\times} \...
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Do homomorphisms $H \to \operatorname{Aut}(K)$ that coincide at the level of $\operatorname{Out}(K)$ induce isomorphic semidirect products?

I recently realized that I don't know of any group that is a nontrivial semidirect product of some symmetric group $S_n$ and another group ($S_n$ being the normal subgroup), except when $n=6$. (For ...
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1answer
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Central extensions versus semidirect products

Consider an extension $E$ of a group $G$ by an abelian group $A$. $$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$ Two special kinds of extensions are: Central Extensions: $A$ is ...
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1answer
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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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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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1answer
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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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1answer
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A necessary condition for two semi-direct products to be isomorphic.

Notations and definitions. For all $A\in\textrm{GL}_n(\mathbb{Z})$, $A$ is hyperbolic if and only if none of its complex eigenvalue has module $1$. For all $(A,B)\in\textrm{GL}_n(\mathbb{Z})^2$, one ...
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Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...
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1answer
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The normaliser of the left regular image [D&F]

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 186): Let $H$ be a group of order $n$, let $K=\text{Aut}(H)$ and form $G=\text{Hol}(H)=H \rtimes K$ (where $...
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On monomial matrices (Generalized Permutation Matrices )

A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices. I have already proved here that the ...
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1answer
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What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
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A necessary condition on semidirect products to be isomorphic

Is it true that the semidirect product groups $\mathbb{Z}\ltimes_A \mathbb{Z}^n$ and $\mathbb{Z}\ltimes_B\mathbb{Z}^n$ are isomorphic if and only if $A$ is conjugate in $\operatorname{GL}(n,\mathbb{...
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1answer
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Isomorphism of semidirect products [D&F]

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6): Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $\varphi_1$ and $\...
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2answers
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Is $G$ always a semidirect product of $[G,G]$ and $G/[G,G]$?

If $G$ is a finite group, it is not true in general that $G$ is the semidirect product of a normal subgroup $N$ and the quotient group $G/N$. It is also not true in general that there is a subgroup ...
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Automorphisms of the affine semilinear group $A\Gamma L(1,2^{n})$

In this question, it is mentionned that the group of automorphisms of the semilinear group $A\Gamma L(1,2^{n})$ is the group itself. Do you have a short proof of this fact?
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1answer
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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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Classification of abelianness-forcing numbers

$n>1$. Prove that $n$ is an abelianness-forcing number iff. $n=p_1^{a_1}p_2^{a_2}\dots p_r^{a_r}$, where $p_1,p_2,\dots,p_r$ are distinct primes, is -cubefree -$p_i\nmid p_j^{a_j}-1,\...
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1answer
110 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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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
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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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1answer
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How is $I(r, n, s)$ a semi-direct product of $H(r, n, s)$ with $C_n$?

I'm due to start my (fully funded!) PhD in Mathematics this October (2017) and I'll be working closely with $H(r, n, s)$, so a detailed answer aimed at that level would be ideal. The Details: ...
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Understanding direct and semi direct products through notations

Can someone please help to understand and identify the following groups? $G_1 = \langle a, b \mid a^{p^2} = b^q = 1, b a b^{-1} = a^i, \operatorname{ord}_{p^2}(i) = q \rangle$. A family of ...
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1answer
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Show that $S_n \cong A_n \rtimes C_2$ [duplicate]

I want to show that $S_n \cong A_n \rtimes C_2$. Take a transposition $\tau \notin A_n$. Then it is clear that $$\langle \tau\rangle \cap A_n = 1$$ $$A_n \tau = S_n$$ $$A_n \unlhd S_n$$ and thus ...
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1answer
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$|G|=pq$, where $p$ and $q$ are primes, is the semidirect product of subgroups of orders $p$ and $q$

I am currently working on the following exercise: Let $G$ be a group of order $pq$, where $p$ and $q$ are primes and $p > q$. Prove that $G = N \rtimes H$ for some subgroups $N$ and $H$ of ...