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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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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Understanding mapping of vertices in Cayley graphs of semidirect products

I came across a small question related to the Cayley graphs of semidirect products of the form $G=(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$. Consider $Cay(G, S_1)$, where $S_1=\{a,b,c\...
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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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120 views

When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
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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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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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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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385 views

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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1answer
115 views

p-Sylow conjugation in semidirect product

I have a semidirect product group $G=N\rtimes H$, where $N$ is a normal $p$-Sylow subgroup of $G$ of order $p^d$ and $H$ has order $m$. Additionally, $m$ and $p$ are coprime and the centralizer of $N$ ...
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59 views

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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187 views

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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236 views

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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49 views

Is there a sort of “two-sided semidirect product”? [duplicate]

Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter. What can we ...
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82 views

What are the semi-direct products of $\mathbb{Z}$ with itself? (Check my work please)

I am just starting out with semi-direct products. I would like to list and describe the semi-direct products of $\mathbb{Z}$ with itself. I first need to find the automorphisms $\varphi$ from $\...
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33 views

Is $\langle A,G\rangle=G\rtimes_{\sigma}A$?

Given groups $A$ and $G$; suppose $A$ act via automorphism on $G$, i.e. there exists $\sigma:A\rightarrow\operatorname{Aut}G$, $a\mapsto\sigma_a:g\mapsto g^a$. From this action we can define a ...
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141 views

If $p$ and $q$ are distinct primes, construct all semidirect products of $\mathbb{Z}_p$ by $\mathbb{Z}_q$.

This is exercise 7.29 in Rotman's book "An Introduction to the Theory of Groups". If $p$ and $q$ are distinct primes, construct all semidirect products of $\mathbb{Z}_p$ by $\mathbb{Z}_q$. I don't ...
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101 views

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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133 views

Find a non-trivial semidirect decomposition of the following groups

Find a non-trivial semidirect decomposition of the groups $S_n$, $n \geq 3$, $D_{2n}$, $n \geq 3$ and $A_4$. Prove that $A_n$, $n \geq 5$ and $Q_8$ have no non-trivial semidirect decompositions. How ...
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1answer
26 views

Find all semidirect products of $(\mathbb{Z}_4,+)$ by $C_2 $

Problem: Find all semidirect products of $(\mathbb{Z}_4,+)$ by $C_2$ (the cyclic group of order $2$). My attempt: We know that $(\mathbb{Z}_4,+)$ is a cyclic group of order $4$. To find all ...
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41 views

Question regarding possiblity for existence of a particular semidirect product

Can there be semidirect products $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$ having $p <q$? I've seen this group for $p>q$ values but not for $p<q$ values, therefore can ...
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76 views

Understanding semidirect product for group of order 30

I am using Dummit and Foote (pg 181-182) and trying to understand section 5.5 on semidirect product for a group of order 30. They start out with a reminder that if $H$ is of order 15 it is a normal ...
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84 views

Semidirect product command in GAP [duplicate]

I need to type the group in GAP. Is it correct if I type as, "SemidirectProduct(ZmodnZ(3),AbelianGroup([7,7]));"? Will it recognize that AbelianGroup([7,7]) is the normal subgroup?
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81 views

Fundamental group of a quotient of a cylinder

Let $X$ be a connected topological space which admits a universal covering. Let $Y$ be the topological space obtained as following:we have an homeomorphism $f : X \to X$ , we take $X \times \left[0,...
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167 views

Interpretation of wreath products in general and on symmetric groups

I've been trying to study how wreath products work. In several textbooks, K wr H is defined as the semidirect product of H acting on the set of all functions from X to K. Now, in my understanding, the ...
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1answer
48 views

Online reference about semi-direct products in finite group theory?

The title says it all: Do you know a comprehensive (preferably online) reference about semi-direct products in (finite) group theory? I would like to know much more about semi-direct products in the ...
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1answer
134 views

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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103 views

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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157 views

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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932 views

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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1answer
57 views

Extension of $\mathbb Z_2$ by $SO(n)$

How to show that the extension of group $\mathbb Z_{2}$ by $\operatorname{SO}(n)$: $$\operatorname{Id} \to \operatorname{SO}(n) \to \operatorname{O}(n) \xrightarrow{\det} \mathbb Z_2 \to 1$$ is a ...
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1answer
234 views

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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176 views

Help with semidirect product

I need help with this problem, i am trying to understand the semidirect product, so if anyine could help or give me some ideas Let $G$ be the group generated by $<a,b>$ and the relations $aba^{-...
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1answer
41 views

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
28 views

Behaviour of restrictions of automorphisms of groups on characteristic subgroup under epimorphisms

Let $G = H \rtimes_\alpha K$, where $H$ is abelian and characteristic in $G$. Let $\phi\in\mathrm{Aut}(G)$, and $\phi'$ is its restriction: $\phi'=\phi\big\rvert_H$. Let $A = B \rtimes_\beta K$, ...
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1answer
55 views

A question regarding groups of order $p^2qr$

When considering finite groups $G$ of order, $|G|=p^2qr$, where $p,q,r$ are distinct primes, let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts faithfully on $...
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1answer
32 views

Identify some Coxeter group

As we all know, the weyl group of lie algebra of $B_{2}$ type is $\left\{s_{1},s_{2}|s_{1}^{2}=1, s_{2}^{2}=1, (s_{1}s_{2})^{4}=1\right\}$. How can we identify this with $Z^{2}_{2}\rtimes S_{2}$? If ...
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1answer
80 views

Semidirect product complements

Suppose that we have a group $A$ containing a normal subgroup $G$ and two complements $H$ and $K$. Symbolically, $H, K \leq A$; $G \unlhd A$; $GH = GK =A$; $G \cap H = G \cap K = \{1\}$; so that $A \...
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81 views

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 ...