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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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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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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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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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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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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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Coxeter presentation of Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$.

I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation $$\langle s_{\text{1}},\ldots,s_n\mid s_{\text{1}}^{\text{2}}=s_i^2=1, (s_1s_2)^4=(s_is_{i+1})^3=(...
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Isomorphisms between semi-direct products

Let $H$ be any group and $K$ an abelian group. (I'm interested in $K={\mathbb Z}$.) Homomorphisms $H\to Aut(K)$ define semi-direct products $K\rtimes H$. There is an action of $Aut(H)\times Aut(K)$ ...
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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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Finding 2 nonabelian, nonisomorphic groups of order 225

I need to find 2 nonabelian, nonisomorphic groups of order 225. Here's what I have so far: Let $G$ be a group of order 225. By Sylow's theorems, we have that $G$ contains $P_{25}$ and $P_{9}$, ...
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Universal covering group of a semidirect product of Lie groups

I have a new relevant question. Let U, V, W be connected Lie groups, so that U is the semidirect product of V and W. By Schreier's theorem (Pontrjagin, Theorem 61), each group has a unique (up to an ...
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Classifying Groups of Order 28

I am trying to classify groups of order 28. In the course of the problem, I am stuck in showing that three semidirect products are isomorphic to each other. In this problem, $G$ is a group of order 28,...
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Classification of Schmidt group to $\mathscr{B}$-group.

Definition: A Schmidt group is defined to be a non-nilpotent finite group with the property that every proper subgroup is nilpotent. Also, a group is said to be a $\mathscr{B}$-group if any ...
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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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“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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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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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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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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Proving $H_1 \rtimes_{\theta_2}K \simeq H_1 \rtimes_{\theta_1}K$

Let $K=C_p$ be a cyclic group of order $p$ (prime). Let $H_1 = C_p \times C_p$, and $\theta_1,\theta_2 : K \to Aut(H_1)$ two homomorphisms. Denote $G_1 = H_1 \rtimes_{\theta_1}K$ and $G_2 = H_1 \...
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Prove that $\exists b\in \Bbb Z$ s.t $x^{ab}=x$.(A semi Direct Product question)

Suppose $K$ is a finite cyclic group, $H$ is an arbitrary group. Consider two homomorphisms $\phi_1, \phi_2: K \to \operatorname{Aut}(H)$ s.t $\phi_1(K), \phi_2(K)$ are conjugate in $\operatorname{Aut}...
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When does a representation of a compact Lie group extend to an extension of the group?

Let $G$ be a compact simple Lie group, let $H$ be its (finite) group of outer automorphisms. I'm interested in the semidirect product $G' = G \rtimes H$. Let $\rho : G \rightarrow U(n)$ be an $n$-...
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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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For any $n ≥3$ there are exactly $4$ distinct homomorphisms from $Z_2$ into $Aut(Z_2^n).$

I tried to prove that for any $n$ $\geq$ 3 there are exactly $4$ distinct homomorphisms from $Z_2$ into $Aut(Z_2^n)$. Also the resulting semidirect products give $4$ non isomorphic groups of order $2^{...
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Normal subgroup $K$ which has no complement.

Rotman's book on Group Theory claims that a normal subgroup $K\trianglelefteq G$ need not have a complement. I recall what is meant by complement. Definition. Let $K$ be a (not necessarily normal) ...
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Role of projective representations in representation theory of semidirect products?

I am interested in the representations of a finite group $G=N\rtimes H$. There is an article by A. Reyes that might be helpful, but I can't find it for free anywhere. This is what I know so far. If $...
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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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Groups of order 306 has 4 types of group containing an element of order 9

There are at most 4 groups of order 306 = $2 \times 3^2 \times 17$, containing an element of order 9. I want to prove the above statement. My trial is below; Let $G$ be such group. If $G$ is ...
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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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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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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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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!
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Wreath product of subgroup with symmetric group

I am going through Generic Quantum Fourier Transforms by Moore et al. I would like to put the screenshot of the section I am confused about below. So, Why does $H$ need to be of size $poly(n)$ for ...
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Direct product, semidirect product and associativity

I know that the direct product operation is associative, and that in general the semidirect product operation is not associative. But what about when we work with a mix of both? I'm trying to ...
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Metaplectic group as semidirect product

There is a homomorphism $$\varphi : \{\pm 1\} \longrightarrow \mathrm{Aut}(SL_2(\mathbb{R}))$$ defined by $\varphi(-1)(A) = (A^{-1})^T$ and this allows us to construct the semidirect product $\{\pm 1\}...
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Conditions for a group to be isomorphic to semidirect product of its subgroups

Let $G$ a group and $N$ a normal subgroup of $G$. If $G$ it have a subgroup $H$ s.t. $H \cap N $ is the trivial subgroup and $H$ is isomorphic to $G/ N$ then $G$ is isomorphic to $N\rtimes H$. Could ...
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Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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How can I show that $D_{2n} \cong C_n \rtimes C_2 $

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
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Is semi-direct product converted to direct product if the normal subgroup is the center of $G$?

Suppose $G$ is a group and $N$ is a normal subgroup in $G$. Also suppose $G=N \rtimes H$. I need to know, is this semi-direct product reduced to the direct product if $N=Z(G)$? My initial guess is ...
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Semi-direct products of $\mathbb Z/12\mathbb Z$ by $\mathbb Z / 2\mathbb Z$

I need to prove that there exists at least three non-isomorphic semi-direct products $$\mathbb Z/12\mathbb Z \rtimes \mathbb Z / 2\mathbb Z$$ To find such semi-direct products, we need to understand ...
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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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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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What is the “bluffer's group” called?

Today I attended a course on Geometric Group Theory in Spanish, and we saw an example of a group which could be literally translated as "bluffer's group", because there is a funny way to interpret it. ...
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Product of two elements in a semidirect product with distinct prime powers

Recall the definition of semidirect product: Let $G, H$ be groups and $\phi:H\longrightarrow \text{Aut}(G)$ a group homomorphism. We define the semidirect product of $G$ and $H$ ($G\rtimes_\phi H$) ...
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Why $\int_{K}f(x,k)\pi_{m}(k^{-1}) dk=g(x)$?

Let $\tilde{\mathbb G}=G \rtimes K$ the semi-direct product of a localement compact group $G$ and a compact group $K$. Let $\pi_{m}$ is a character of the irreducible unitary representations of $K$. ...
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Show that $D_3\times_\rho\mathbb{Z}_2$ is not isomorphic to $A_4$

This is the third part of a problem. I will list here the first two parts as a reference and then my attempt to solve the third one. I want to verify if the first and second are right and some help ...
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Representations of the generalized quaternions

Given the generalized quaternion group $Q_{2^t} = \langle a,b : a^{2^{t-1} } = e, b^2 = a^{2^{t-2}}, b^{-1}ab = a^{-1}\rangle$, I have been asked to find it's (linear complex) representations. My ...
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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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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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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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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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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 ...