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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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0answers
208 views

Understanding torsors and semidirect products of groups

I'm trying to understand the semi-direct product of groups from either a categorical or a geometric perspective and failing miserably. The four things I'm hoping will fit into a coherent picture are: ...
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Can this matrix group be obtained from $(\mathbb R,+)$ and $(\mathbb R^*,\cdot)$?

I have stumbled upon this group of matrices in an old midterm: $$G=\{ \begin{pmatrix} a & b \\ 0 & \frac1a \\ \end{pmatrix}; a,b\in\mathbb R, a\ne0 \}.$$ The students were asked to show ...
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Visualizing $S_3 \rtimes D_4$

I am trying to visualizing $S_3 \rtimes D_4$ following this video. Here, $S_3$ is the symmetric group over three symbols and $D_4$ is the dihedral group of order $8$. The semidirect product is defined ...
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1answer
377 views

Inner vs outer semidirect products of $S_3$ and $D_4$

I am trying to understand the difference between the inner and outer semidirect products of the symmetric group $S_3$ and the dihedral group $D_4$ of order $8$. The products are defined here. Inner ...
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1answer
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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 \...
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Computing the characters of $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $

Let a group be $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $ where $t \in \mathbb{Z} \text{ and } t \ge 1; N_i, m_i \in \mathbb{Z} \text{ and } N_i, m_i \ge 1$; and $S_{N_i}$ is a symmetric group over $N_i$ ...
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1answer
731 views

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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What is $Z(\mathbb{Z}_{p} \rtimes_{\phi} \mathbb{Z}_{q})$ if $\phi$ is non-trivial and $p$ and $q$ are primes such that $q \equiv 1(\text{mod} \ p)$?

Let $p$ and $q$ be prime numbers such that $q \equiv 1(\text{mod} \ p)$. It is known that a group of order $p \cdot q$ is either isomorphic to $$\mathbb{Z}_{p} \times \mathbb{Z}_{q} \cong \mathbb{Z}_{...
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Classify all groups of order 3825

I am trying to classify all groups of order $3825=3^2 \cdot 5^2 \cdot 17$. The Sylow theorems indicate that the number of Sylow p-subgroups for each p rime are $n_{17}=1$, and $n_{3}=1,25,85$ and $...
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Why $W(B_\ell) \simeq (\Bbb Z/2\Bbb Z)^\ell \rtimes S_\ell$?

Let $V:=\Bbb R^\ell $, $\ell \geq 2$ with an orthonormal basis $\{e_i\}_{i=1}^\ell $. The set $$\Phi = \{ \pm e_i \pm e_j \mid 1 \le i\neq j \le \ell\} \cup \{ \pm e_i \mid 1 \le i \le \ell\} $$ is ...
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1answer
81 views

Is this a special semidirect product?

Currently I am going through Representation theory of semi-direct products by Reyes. At the begining of the article, the author defines the semidirect product as follows. $G = H \cdot B$ is a ...
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241 views

Characters of semidirect and wreath products of non-Abelian finite groups

In Section 3.2 of Linear Representations of Finite Groups by Serre, the author gives a formula to compute the characters of product of finite groups (Abelian or non-Abelian). First Serre defines the ...
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1answer
481 views

Split extension of groups and semidirect product

I am studing Ext functor and have some basic problem. For every semidirect product $G$ of groups $N$ and $H$, short exact sequence $0 \to N \to G \to H \to 0$ splits. On the other hand, every ...
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Computing $\operatorname{Aut}(D_{2n})$ and realizing it as $\mathbb{Z}/n\mathbb{Z} \rtimes_{\psi} (\mathbb{Z}/n\mathbb{Z})^{\times}$

I have no idea how to approach this problem. I would like to find $\psi$ that does this. Given the fact that we know the presentation $D_{2n} = \langle x,y : x^n = e, y^2 = e,yxyx = e \rangle$.
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1answer
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Do (pseudo)varieties closed under wreath product have a name?

Pseudovarieties of finite monoids are often studied in conjunction with the class of regular languages they recognize (a monoid $M$ recognizes a language $L \subseteq A^*$ if $L = h^{-1}(h(L))$ for ...
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Anything wrong with this proof on Rubik's cube group?

I'm writing my thesis on semidirect products. As an example, I am proving that the Rubik's cube group $G$ is the semidirect product of its orientation-preserving moves subgroup $C_P$ and position-...
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223 views

Difference between internal and external semi direct product

Currently i am trying to understand the concept of Semi direct product of groups from Abstract Algebra text of Dummit and Foote.The discussion given in the same book is bit confusing to me and i am ...
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2answers
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Prove that the Dihedral group $D_n$ is isomorphic to $Z_n \rtimes_{\psi} Z_2$

I consider the following map $\psi : Z_2 \rightarrow Aut(Z_n)$ where we map the identity element 0 to the identity map and $1 \mapsto \theta : Z_n \rightarrow Z_n$ where $\theta(x) = -x$. I am not ...
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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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Problem with an example of semidirect product

I was trying to understand Example 7.15 from Rotman's Introduction to the Theory of Groups. The problem I'm having is maybe a bit silly and not strictly about semidirect products, but I don't ...
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Commutative diagram of semidirect products

My question is partly motivated by trying to solve this one. Let $E$ be the semidirect product of groups $G$ and $H$. Then, we have an exact sequence: $$G \hookrightarrow^\iota G \rtimes_\phi H \...
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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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1answer
192 views

Split short exact sequence equivalent to internal semidirect product

Given a short exact sequence $1 \to N \to G \to H \to 1$, $\alpha: N \to G, \beta: G \to H$ which is split: there exists $s: H \to G$, s.t. $\beta \circ s = \text{id}_H$, I want to see that $G$ is the ...
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1answer
187 views

semidirect product of isometry group

I am doing exercise about semidirect product. Here is the question: Prove that the isometry group of Euclidean space $R^n$ is $O(n)\rtimes R^n$. I was stucking. Any ideas?
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Finding all elements of order 7 in a semi-direct product group

Let $H=<h>,\ G=<g>,\ o(h)=7,\ o(g)=3$, and let $\alpha:G\rightarrow Aut(H)$ so that $\alpha(g)(h) = h^2$. Find all the elements of order 7 in $H\rtimes_\alpha G$. We get that $\alpha(g^j)(...
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on the properties of semidirect product of finite groups

If $G$ is a finite group such that $G/H \cong K$, where $H,K \leq G$, then is it true that $G \cong H \rtimes K$? it is sufficient to show that $H \cap K=1$ and $G=HK$.
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I don't quite understand the definition of wreath product

In Rotman's group theory book, a wreath product is defined as $$D\wr_\Omega Q := K \rtimes Q$$ where $\rtimes$ is the semidirect product, $K = \prod_{\omega \in \Omega} D_\omega$, and $D_\omega \...
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70 views

Semidirect product with isomorphic copies of complements

The definition of semidirect group on Rotman's Introduction to the theory of groups is the following: A group $G$ is a semidirect product of $K$ by $Q$, denoted by $G=K \rtimes Q$, if $K \lhd G$ 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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1answer
121 views

About Properties of Semi-direct Product

I'm currently studying semi-direct product of groups (from the very famous book from Dummit & Foote). This is the context (page 176): Let $H$ and $K$ be groups, $\varphi\in Hom(K,Aut(H))$. Let $\...
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63 views

Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$

Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Lets say the generators of ...
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1answer
209 views

Subgroups of a semidirect product with a cyclic group of order $2$

Let $G$ be a finite group. Is it true that every subgroup of $G\rtimes_{\varphi} \mathbb{Z}/2\mathbb{Z}$ is of the form $H$ or $H\rtimes_{\varphi} \mathbb{Z}/2\mathbb{Z}$, where $H\subset G$ is a ...
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Extension of elementary abelian p-groups

Let $H$ and $K$ be groups. An extension of $K$ by $H$ is a triple $(i,G,p)$ where $G$ is a group, $i:H\longrightarrow G$ is an injective homomorphism and $p:G\longrightarrow K$ is a surjective ...
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412 views

Order of elements in Semidirect product of groups

Let $G$ and $H$ be two finite groups, where $H$ acts on $G$ trivially, so that $\eta_h(g)=g$ for all $g\in G$ and $h\in H$, and $G$ acts on $H$ by conjugation. We want to construct the semi direct ...
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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 there only one internal semidirect group?

I am a bit confused on the concept of internal semidirect group $N\rtimes H$. For external semidirect group $N\rtimes_\varphi H$, I know that there can be possibly many semidirect groups depending on ...
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106 views

How to write $\Bbb Z_q\rtimes\Bbb Z_p$ as $\langle a,b\mid a^p=b^q=1,aba^{-1}=b^{i_0}\rangle?$

I am referring specifically to this example http://planetmath.org/groupsoforderpq In Case 2, the group should be $\mathbb{Z}_q\rtimes\mathbb{Z}_p$. How do we write it in the presentation of $$G=\...
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1answer
221 views

Left vs right semi direct products

I just want to make sure that I am not doing anything silly here, but if we let $G$ be a group with $H,K$ subgroups, $H\lhd G$, and $\phi:K\rightarrow Aut(H)$, then is $$H\rtimes_\phi K \approx K \ _\...
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Finding a 3-embedded subgroup.

I have the group of order $108$, $$G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$$ obtained from an algorithm in GAP, but I need to prove that ...
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135 views

An example of semi direct product of Lie algebras

Let $L$ be Lie algebra and $d: L \to L $ a derivation. Let we have a one-dimensional Lie Algebra generated by element $t$. What will be the semi direct product of $L$ with one-dimensional Lie algebra ...
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1answer
77 views

Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $ U $ on the diagonal. Find a Sylow-p-...
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1answer
116 views

Subgroup of semidirect product

Let $G$ be a semidirect product of a normal subgroup $A$ with a subgroup $B$ and Let $H$ be a subgroup of $G$ such that $H\cap A$ is trivial. Is it true that $H$ is contained in a conjugate of $B$ ? ...
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160 views

External Semidirect product and isomorphism

Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism. Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ ...
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63 views

problem about inner semidirect product

Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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110 views

subgroup of a semidirect product

I'm really lost with this problem and I really need your help: Let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$, and let $H\leq G$ with finite index in G. I have to prove that there is a subgroup $U$ of $\...
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3answers
368 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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1answer
247 views

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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1answer
355 views

Center of a semidirect product

Here http://planetmath.org/node/87994 a formula for the center of the semidirect product of two groups for a given homomorphism is given. I also wonder whether the formula is correct or not. The ...
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1answer
140 views

Understanding a group of order $2^{25}.97^2$

Let $G$ be a semidirect product of a Sylow 2-subgroup $P$ and a normal subgroup $Q$. $P$ is itself is semidirect product as defined below: $$P=(\langle u \rangle \times \langle v \rangle \...
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31 views

Is it true that if $bk\in Z(B\rtimes K)$ then $b\in Z(B)$?

Is this fact true? Let be $G= B \rtimes K$ and I suppose that $w=bk \in Z(G)$, with $b\in B$ and $k \in K$. Is it then true that $b\in Z(B)$? I have tried to prove it but I didn't succeed. Thanks!