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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242 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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136 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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138 views

Homomorphisms $H \to \operatorname{Aut}(K)$ that induce isomorphic semidirect products for centerless $K$

This is a followup to this previous question: Do homomorphisms $H \to \operatorname{Aut}(K)$ that coincide at the level of $\operatorname{Out}(K)$ induce isomorphic semidirect products? I am trying ...
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49 views

Semidirect product when the action factors

Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors ...
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188 views

Galois Group of the splitting field of the polynomial of $x^{11} - 7$ over $\mathbb{Q}$

Be $\mathbb{L}$ the splitting field, and be $G$ the Galois Group of $\mathbb{L}/\mathbb{Q}$, I've to prove that $G \cong A \ltimes B $ with $|A| = 11$ (cyclic) and $|B|=10$, abelian. The central ...
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304 views

Showing a Group $G$ is a Semidirect Product of $S_n$ and the Group of Diagonal Matrices with Entries $±1$.

Consider $G$ to be the set of $n$ $\times$ $n$ matrices with entries in $\{\pm1\}$ that have exactly one nonzero entry in each row and column. These are called signed permutation matrices. Show that ...
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171 views

Prove that $C_H(K) = N_H(K)$ for $G=H \rtimes_{\phi} K$

Let $H,K$ be group where $\phi: K \rightarrow \operatorname{Aut}(H)$ is a homomorphism. Also, let $G=H \rtimes_{\phi} K$. Show that $C_H(K) = N_H(K)$ Proof: Let $h\in N_H(K)=\{h\in H: hKh^{-1}=K\}$...
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81 views

Identifying groups with subgroups isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$

I was playing around with semidirect products and tried finding a non abelian semi direct product of $\mathbb{Z}_2\times \mathbb{Z}_2\rtimes \mathbb{Z}_2$. I couldn't find a group that worked, and I ...
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52 views

When $\mathbb{Z}^n \rtimes \mathbb{Z}_m$ nilpotent?

For what values of $m,n$, and what kind of actions by $\mathbb{Z}_m$ , $\mathbb{Z}^n \rtimes \mathbb{Z}_m$ nilpotent?
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99 views

Build a semidirect product $\mathbb{Z}/p^2\mathbb{Z}\rtimes \mathbb{Z}/p\mathbb{Z}$

For $p$ prime and odd I'm trying to build a non trivial semidirect group $\mathbb{Z}/p^2\mathbb{Z}\rtimes \mathbb{Z}/p\mathbb{Z}$. So, for that, I look for the homomorphisms between $Aut\left(\...
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276 views

semidirect product of matrix group in GAP

I am new in gap, please accept my apologize because of asking some simple questions. I want to know if we have two general linear groups, is it possible to make the semidirect product of them in gap. ...
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58 views

On subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$ [closed]

In order to study the subgroup structure of the p-group $G\simeq (Z/pZ)^{m}\rtimes(Z/pZ)$, I need to solve the following exercice from the Book (Dummit & Foote p101): Exercice: Let $H$ be a ...
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For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly.

For all $n > 0$ express $D_{2n}$ as a semidirect product $\mathbb Z_n \rtimes_\theta \mathbb Z_2$, finding $\theta$ explicitly. I am not sure how to go about finding $\theta: \mathbb Z_2 \to \...
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33 views

Find all group of order $20$ which is a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$

Problem: Find all group of order $20$ that are a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$. My attempt: We knew that a cyclic group of order $n$ is isomorphic ...
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124 views

Prove that for the semidirect product $\mathbb{Z}_5 \rtimes \mathbb{Z}_3$, the homomorphism $\alpha$ is trivial

Suppose $G \cong \mathbb{Z}_5 \rtimes_\alpha \mathbb{Z}_3$ with respect to a homomorphism $\alpha:\mathbb{Z}_3 \to \mathrm{Aut}(\mathbb{Z}_5)$. Show that $\alpha$ is trivial and that $G \cong \mathbb{...
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Maximal normal locally nilpotent subgroup in an infinite extension of a Prüfer group

Let $P$ be a Prüfer $p$-group, let $x$ be an infinite automorphism of $P$, namely a $p$-adic unit, and consider the semidirect product $G$ between $P$ and $\langle x\rangle$ via the action of $x$ on $...
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44 views

How is $I(r, n, s)$ a semi-direct product of $H(r, n, s)$ with $C_n$?

I'm due to start my (fully funded!) PhD in Mathematics this October (2017) and I'll be working closely with $H(r, n, s)$, so a detailed answer aimed at that level would be ideal. The Details: ...
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225 views

Prove that $C_K(H) = \ker \varphi$

I'm doing an exercise in Dummit and Foote's Abstract Algebra. Here's the setup; Let $H, K$ be groups, let $\varphi:K \to \operatorname{Aut}(H)$ be a homomorphism, and identify $H, K$ as subgroups ...
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69 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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103 views

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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167 views

$|G|=pq$, 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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119 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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340 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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288 views

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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19 views

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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338 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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121 views

How many non isomorphic semidirect products are there between $\mathbb Z_2$ and $SL(2,3)$?

I know that $GL(2,3)$ is one of this, but i need the characterization of all possibles of the semidirect products between $\mathbb Z_2$ and $SL(2,3)$. Thanks, for any help.
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248 views

Semidirect Product Equivalent Conditions

First, I am sorry if this post has been posted here before since I cannot find anything related to it. I read on wikipedia about the equivalent conditions for semidirect product http://en.wikipedia....
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68 views

How do I compute the normaliser of a group G, considered as a set, in the group of set bijections of G?

Suppose G is a group and T(G) is the group of set bijections of G. I identify the elements of G as maps corresponding to the left multiplication by the chosen element. Then the normaliser of G in T(G) ...
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276 views

External semidirect product application

I am trying to find all normal subgroups of $\mathbb D_n$. I've read here Normal subgroups of dihedral groups that one could show the external semidirect product $(\mathbb Z/n\mathbb Z) \rtimes (\...
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Question regarding writing a group of order $p^2qr$ using notations

Let $G$ be a solvable, non-nilpotent group of order $p^2qr$, where $p,q,r$ are distinct primes, and let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts ...
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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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67 views

Prove that $S_4 \cong V_4 \rtimes_\phi S_3$ for any isomorphism $\phi: S_3 \to \text{Aut}(V_4)$

Note that $\text{Aut}(V_4) \cong S_3$. I know how to prove that $S_4$ isomorphic to some semidirect product of $V_4$ and $S_3$. I know if it works for an isomphorism it works for any isomorphism. ...
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Are these two semidirect products isomorphic?

Let $p$ be a prime. Then there is a nonabelian semidirect product of $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p$. There is also a nonabelian semidirect product of $\mathbb{Z}_p\oplus\mathbb{Z}_p$ and $\...
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56 views

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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An upper bound for the number of semidirect products of a finite group up to isomorphism

Given a group $G$ with identity element 1, a subgroup $H$, and a normal subgroup $N$ of $G$; $G$ is called the semidirect product of $N$ and $H$, written $G = N\rtimes H$ , if $G = NH$ and $H\cap N=1$...
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50 views

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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Can anyone help me prove that the following statement implies G finite group is directly indecomposable? [closed]

Statement: If for any K1 and K2 groups, K1×K2 has an isomorphic subgroup to G, then K1 has a subgroup isomorphic to G and K2 also has a subgroup isomorphic to G. I know that an example when this ...
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139 views

Determine all homomorphisms $\phi: K\rightarrow Aut(H)$.

Let $H$ be cyclic group of order 7 and let $K$ be cyclic of order 3. (a) Determine all homomorphisms $\phi: K\rightarrow Aut(H)$. My attempt: Since $K$ is cyclic, any one of the two generator of $K$ ...
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51 views

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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58 views

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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89 views

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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38 views

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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24 views

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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55 views

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 ...