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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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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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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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Intuition about the semidirect product of groups

If we have two groups $G,H$ the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product $G\times H$ into a group it is certainly by ...
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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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What is the history of the semidirect product?

It's not hard to imagine early group theorists getting the inspiration for the semidirect product because after you've seen a few examples of finite nonabelian groups, the pattern starts to emerge on ...
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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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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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Semidirect product of two cyclic groups

Describe all semidirect products of $C_n$ by $C_m$ (ie $C_n \rtimes C_m$) where $m,n \in \mathbb{N_+}$ Note: For the first attempt one needs to find all homomorphisms from $C_m \to U(n)$, but the ...
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Does $\varphi_1(K) \cong \varphi_2(K)$ imply $H\rtimes_{\varphi_{1}} K \cong H\rtimes_{\varphi_{2}} K$?

I recently found myself trying to prove (or disprove) the following lemma: Lemma: Let H, K be groups and let $\varphi_1, \varphi_2 \colon K \rightarrow \mathrm{Aut}(H)$ be homomorphisms. Suppose also ...
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Is a finite centerless metabelian group always a semidirect product of two abelian groups?

Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups? It does not seem true to me, but I failed to find any counterexamples. Actually, ...
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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 $\operatorname{Z}(...
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What's the smallest non-$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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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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Non cyclic group of order $p^3$ satisfies $G \simeq H \rtimes_{\theta}K$

Let $G$ be a non-cyclic group of order $p^3$ for an odd prime $p$. Prove that $G \simeq H \rtimes_{\theta}K$, where $H$ is a normal subgroup of $G$ of order $p^2$, $K$ is a subgroup of order $p$, and $...
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Intuition on the external Zappa–Szép product

$\newcommand{\Aut}{\operatorname{Aut}}$A classmate of mine recently posted an interesting question on Facebook. It didn't get an answer, and I couldn't get anywhere myself, so I'm hoping that someone ...
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How can I check whether a given finite group is a semidirect product of proper subgroups?

Suppose, a finite group $G$ is given. I want to check whether there is a proper normal subgroup $N$ of $G$ and a subgroup $H$ of $G$, such that $G$ is the semidirect product of the groups $N$ and $H$...
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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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Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
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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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Is $S_4 \times C_2$ isomorphic to $(C_2 \times C_2 \times C_2) \rtimes S_3$

Let $S_n$ denote the symmetric group on $n$ letters and $C_n$ denote the cyclic group of order $n$. Consider $(C_2 \times C_2 \times C_2) \rtimes S_3$ where $S_3$ acts on $(g_1, g_2, g_3) \in C_2 \...
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Group of order 7.11.17 is cyclic

Let $G$ be a group of order $7.11.17$. Show that $G$ is cyclic. I tried to find a solution using Sylow theorems but I got stuck, here it goes: We know that $$n_7 \equiv 1 (7) \space, n_7|11.17 \...
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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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Classifying groups of order 585

I am trying to classify the groups of order 585. (It is known that there are 4 of distinct non-isomorphic groups, but I am not assuming it.) The question further asks to show that any group of this ...
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1answer
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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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$(\mathbb Z/p \mathbb Z \rtimes \mathbb Z/q \mathbb Z) \times \mathbb Z/q \mathbb Z \cong\mathbb Z/p \mathbb Z \rtimes (\mathbb Z/q \mathbb Z)^2$?

Given: Let $p$ and $q$ be prime numbers such that $q$ divides $p-1$. It is well-know that there is a monomorphism $\varphi: \mathbb Z/q \mathbb Z \to Aut(\mathbb Z/p \mathbb Z)$. Define ...
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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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Semidirect product: general automorphism always results in a conjugation

When $G$ is a group, $N$ is a normal subgroup of $G$ and $H$ is another subgroup of $G$ where $ N \cap H = \{1\} $, the normality of $N$ suggests that we can write, for $n_1, n_2 \in N$ and $h_1, h_2 \...
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The semidirect product $(C_7\times C_{13})\rtimes C_3$

Following this page, in the classification of groups of order $273$, the product of the Sylow group $S:=C_7C_{13}\simeq C_{7}\times C_{13}$ is normal, hence can be acted on by the Sylow $C_3$ to ...
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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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Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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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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Projection of a normal subgroup in semidirect product

Consider a semidirect product $N\rtimes G$. Consider the projection map $\pi_N\colon N\rtimes G\to N$. Suppose $\Gamma\unlhd N\rtimes G$ is a normal subgroup and that $\pi_N[\Gamma]=H$ is a subgroup ...
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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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Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in GL_k(\mathbb{R}),\;...
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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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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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Affine group, semi-direct product and linear transformations

According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
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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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What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
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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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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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Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, \ldots,...
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Semi-direct product in general linear groups

$\operatorname{GL}(n,F)$ can be written as a semidirect product : $\operatorname{GL}(n,F) = \operatorname{SL}(n,F) ⋊ F^\times$ where $F^\times$ is multiplicative group of the field $F$. According to ...
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What are differences between semidirect product and direct product?

Given two groups $A, B$, we can construct direct product $A \times B$ whose elements are of the form $(a, b), a \in A, b\in B$. If $A, B$ are subgroups of a group $G$ and $A \cap B =\{1\}$, then we ...
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Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$

When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following: $\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$ See https://groupprops....
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How many non-abelian groups of order $lpq$ are there?

If $l,p,q$ are primes with $l<p<q$, such that $$p\nmid (q-1)\hspace{1cm} l\mid (p-1)\hspace{1cm} l\mid (q-1) $$ I want to show that there are at least $1$ and at most $(l+1)$ non-...
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1answer
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Why is the symmetry group $S_3$ not the direct product of two nontrivial groups?

I know that $S_3$ is the semidirect product of $\bigl\langle(1\ \ 2\ \ 3)\bigr\rangle \rtimes\bigl\langle(1\ \ 2)\bigr\rangle$, and I'm not sure where exactly the direct product property fails. Is it ...
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Proof for: semidirect product of solvable groups is solvable

Do you know the proof for the following statement or where I can find it? Semidirect product of solvable groups is solvable? I thought that this property is so ...