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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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72 views

Find an explicit isomorphism between $H\rtimes_{\phi} K$ and $S_3$.

So this is a past paper question that I am currently struggling with: Suppose $H \cong C_2$ and $K \cong C_3$, and $\phi: H \rightarrow Aut(K)$ is non-trivial. Find an explicit isomorphism between $H ...
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Free group as semi-direct product

Let $F$ be a free group of rank at least $2$. Of course then, $F$ can not be direct product of two subgroups (except the trivial decomposition $1\times F$). Q. Can $F$ be written as semi-direct ...
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Showing $\Bbb{Z}_q \rtimes Q_8$ has the presentation $\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$

Let $G \cong \Bbb{Z}_q \rtimes Q_8$. Then $G$ has a presentation as follows $$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$$ I dont understand why $x^z=x^{-1}$? ...
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Can a factorisation be pulled out of a semidirect product in this case?

I want to know if my proof of the following is true, since I am not proficient enough in group theory to be sure by myself. Proposition Let $G,H$ be groups, $N\triangleleft G$. Let there be a ...
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98 views

Semi direct product $G \cong N \rtimes_{\varphi} K $

So Im having some hard time in trying to understand the semi direct product. So I get that given two groups $N$ and $K$ and a homomorphism $\varphi:K \to Aut(N)$ we can define the semi direct product $...
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134 views

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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77 views

Semidirect product with subgroup

Let $G,H$ be groups with an action of $H$ on $G$, meaning a group homomorphism $H\to Aut(G)$, and let $S\le H$ be a subgroup, not necessarily normal. Consider the semidirect products $G \rtimes H$ ...
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Represent a group in the form of generator and relation

The only nonabelian semidirect product of $\mathbb Z\rtimes_{\theta}\mathbb Z$ is when $\theta(1)=-1$. And write it in formula, we have $$(a,b)(c,d)=(a(-1)^{b}c,bd).$$ But I can't find a good way to ...
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A question on classification of groups of order 30

I was reading the classification of groups of order 30 from Dummit & Foote(pg-182, 3rd ed). It has a normal subgroup of order 15 which is obviouly cyclic. Hence G is isomorphic to $\mathbb{Z}_{15} ...
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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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Help understanding/proving : E(n) = O(n) ⋉ $\mathbb{R}^n$

I have been reading through some book, such as Geometry of Crystallographic groups (by Andrzej Szczepanski), and during this reading I came across the relationship: E(n) = O(n) ⋉ $\mathbb{R}^n$. ...
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Proof that $G$ is the semi direct product of $P$ and $Q$ if and only if the composition $\phi\circ \iota : P\rightarrow G/Q$ is an isomorphism

Let $G$ be a group with subgroups $P$ and $Q$. Assume $Q$ is normal. Define $G$ to be the semi direct product of $P$ and $Q$ if $PQ=\{pq: p\in P,q\in Q\}=G$ and $P\cap Q=\{e\}$, where $e$ is the ...
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Smallest finite group larger than the Quaternion group that isn't an outer Semi Direct Product

Define the outer-semi direct product in the standard way https://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products I found out that $Q_8$ is the smallest finite group that is ...
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Semidirect product structure on orthogonal group

Using that the orthogonal group $O(2n)$ consists of two connected components, which are $SO(2n)$ and $RSO(2n)$ (where $R=\operatorname{diag}(-1,1\ldots,1)$), one can construct a diffeomorphism $\phi:O(...
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Conditions on $\alpha, \beta$ under which $A \rtimes_{\alpha} B $ and $A \rtimes_{\beta} B$ are isomorphic

Let $Q$ and $N$ be two groups. If there are two homomorphisms $\alpha,\beta : Q \rightarrow \operatorname{Aut}(N)$ then we can construct the semidirect products $G_a = N \rtimes_{\alpha} Q $ and $G_b =...
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$|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 ...
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265 views

Semidirect product of groups by magma

Can anybody guide me towards, how I can compute semidirect product of $\mathrm{PSL}(3,4)$ and $\mathbb Z_2$ by magma? Indeed, I dont know how construct map $\phi: H \to \mathrm{Aut}(N)$, when $H=\...
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Faithful Action of N $\rtimes$ G on N

In short, I am trying to find a faithful action of $N \rtimes G$ on $N$, where I know that the action for the semidirect product is faithful. My first attempt was $(n, g) \cdot n'=(nn') \cdot g$, but ...
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Complement of the kernel of a group epimorphism of a semidirect product

Currently, I am going through Chapter 25 of Character theory of finite groups by Bertram Huppert. Here the authors introduce the concept of semidirect product $G \wr H$ and the kernel of its ...
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What does the bar mean in $M = G \wr H = B \bar{H}$?

Currently, I am going through Chapter 25 of Character theory of finite groups by Bertram Huppert. On page $343$, there is a lemma on semidirect products. It starts as follows. 25.5 Lemma. Let $M = ...
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Why is the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ $16$?

Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ is $16$. My ...
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Smallest symmetric group which embeds $D_n \rtimes Aut(D_n)$

This question answers which the smallest symmetric group to embeds $D_n$ of order $2 n$. I would like to know what is the smallest symmetric group which embeds $D_n \rtimes Aut(D_n)$. I understand ...
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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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146 views

Are Etingof and Serre the only books to discuss representation theory of semidirect product?

I have found discussions about the representation theory of semidirect products in the Section 4.26 of Introduction to Representation Theory by Etingof et al and Section 8.2 of Linear Representations ...
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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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Representations of semidirect products of non-abelian groups

I am trying to define representations of semidirect products of non-abelian groups following the example of representations of semidirect products of abelian groups by Etingof et al. Here is what I ...
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125 views

Find irreducible representations of semidirect product $(S_2 \times S_2) \rtimes S_2$

I'm looking at a very specific action of the semidirect product (wreath product) $(S_2 \times S_2) \rtimes S_2 = S_2 \wr S_2$ on $\mathbb{Q}^3$. Namely, the generators acts as follows on a basis $...
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166 views

Isomorphism between $HK$ and $H⋊ K$

Consider the semidirect product $H⋊ K$ of the groups $H$ and $K$. We know that given a group $G$ such that $H,K\leq G$, we can recognize that it is the semidirect of $H$ and $K$ provided that (i) $H\...
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SemiDirect Products-Recognizing when they are isomorphic or not

I know there are methods of showing when $H\rtimes_{\Psi_{1}}K$$\cong$$H\rtimes_{\Psi_{2}} K$. However, what about semidirect products in which the H's differ. Is it ever the case where $|H_{1}|=|...
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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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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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Classifying groups such that $G \simeq \mathbb{Z}_3 \rtimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$

I am thinking G will be $D_{12}$; however, I am not sure how to prove that all semi-direct product are isomorphic or explicitly get $D_{12}$. We have $\mathbb{Z}_2\times \mathbb{Z}_2$ has two ...
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Is semi-direct product represent all group (up to isom) of order 6?

We have $|G|=6$ and by sylow thm we have $n_{3}=1\ n_{2}=1\ or\ 3$ this implies $\exists H \in Syl_{3}(G)$ and $H\lhd G$. Let $K \in Syl_{2}(G)$, then $H\cap K = {1}$ and $HK=G$ and $G\cong K\times_{\...
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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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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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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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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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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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238 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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473 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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59 views

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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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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142 views

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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214 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 ...