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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Number of Subgroups in $(\mathbb Z_8 \times \mathbb Z_2) \rtimes \mathbb Z_2$

In the $2$-group $G$ of order $32$ which has Id$ [32,5]$, and could be describe as: $$\begin{align} G &\cong \langle {a,b,x:a^8,b^2,x^2,[a,b]=[a,x], ...} \rangle \\ &\cong (\mathbb Z_8 \...
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Galois group of $L=\mathbb{Q}(i,\sqrt[3]2,\sqrt3)$ over $\mathbb{Q}$ is $D_{12}$

Consider $L = \mathbb{Q}(i,\sqrt[3]2,\sqrt3)$. Prove that $\operatorname{Gal}(L/\mathbb{Q})\cong D_{12}$. My attempt: It is easy to verify that $[L:\mathbb{Q}]=12$. In particular, $L$ is the ...
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Different constructions of $(\Bbb Z_4 \times \Bbb Z_2) \rtimes\Bbb Z_2$

I read that there are only two semidirect products $(\Bbb Z_4 \times\Bbb Z_2) \rtimes\Bbb Z_2$ whose presentations are given by \begin{align} (\Bbb Z_4 \times\Bbb Z_2) \rtimes\Bbb Z_2 : \langle a, b, ...
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Properties of functions $f(g,\beta\alpha,k+n)=f(g^\alpha,\beta,k)f(g,\alpha,n)$ and outer semi-direct products.

Given a two groups $G, N$ with $N$ possibly abelian. Let $g,\alpha,\beta\in G$ and $x,y\in N$. I'm interested in studying functions $f:G\times G\times N\to G$ that satisfy this functional equation (...
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Nilpotent quotient of semidirect product of a nilpotent group and a free abelian group

Let $N$ be a finitely generated infinite nilpotent group and let us denote by $G$ the semidirect product $N \rtimes \mathbb{Z}^n$ for some $n\in \mathbb{N}$. I would like to know if there is an ...
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Let $H=\langle\sigma\rangle$ be the infinite cyclic group. Consider $G = (\mathbb{Z}\times \mathbb{Z})\rtimes H$. Determine $G'$.

Let $H=\langle\sigma\rangle$ be the infinite cyclic group. Consider $G = (\mathbb{Z}\times \mathbb{Z})\rtimes H$ with action given by $$ \sigma:\mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}\times\mathbb{Z}:...
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Is a holomorph of a group unique?

Suppose $G$ is a group. Then, a holomorph of $G$ is defined as a outer semidirect product ${\rm Hol}(G) = G \rtimes_{\phi} {\rm Aut}(G)$. Since an outer semidirect product depends on a automorphism $\...
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irreducible representations and characer table of $G = \mathbb{Z}/4\mathbb{Z} \times ( \mathbb{Z}/25\mathbb{Z} \ltimes \mathbb{Z}/4\mathbb{Z})$

What are the irreducible representations and characters of this group $G = \mathbb{Z}/4\mathbb{Z} \times ( \mathbb{Z}/25\mathbb{Z} \ltimes \mathbb{Z}/4\mathbb{Z})$ of order $|G| = 400$ ? Perhaps we ...
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Is the nontrivial semidirect product of $C_3^2$ by $Q$ unique?

Is the nontrivial semidirect product of $C_3^2$ by $Q$ unique? This is an extension of a homework problem about group representations, but some of the things I can extract from the group ...
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$\Bbb{Z}_{26} \rtimes_{\alpha} \Bbb{Z}_{5} \cong \Bbb{Z}_{26} \times \Bbb{Z}_{5}$ as groups.

If $\alpha: \Bbb{Z}_{5} \rightarrow \text{Aut}(\Bbb{Z}_{26})$ is some group homomorphism, how do we show that $\Bbb{Z}_{26} \rtimes_{\alpha} \Bbb{Z}_{5} \cong \Bbb{Z}_{26} \times \Bbb{Z}_{5}$ as ...
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How do I show a semidirect product of relatively-prime ordered groups is isomorphic to the direct product of those groups?

Here is my question: Given that $a$ and $b$ are relatively prime positive integers, and $\alpha$ is some group homomorphism $\alpha: \Bbb{Z}_{b} \rightarrow \text{Aut}(\Bbb{Z}_{a})$, where $\text{Aut}(...
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Semidirect products of 3 groups?

When we have a group $A$ acting on a group $B$ it is possible to construct a third group "containing" both $A$ and $B$ where the action of $A$ on $B$ coincids with the conjugation of the ...
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Let $(n,h)\in G = N \rtimes H$. Is it true that $\mathrm{ord}(n,h)=\mathrm{lcm}(\mathrm{ord}(n),\mathrm{ord}(h))$? [closed]

Let $G$ be a group which is the (outer) semi-direct product of groups $N\rtimes H$. $n\in N,h\in H$. Is that true that the order of $(n,h)$ in $G$ is equal to the lcm of $\mathrm{ord}(n)$ and $\mathrm{...
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Can this group be written as a semidirect product?

Assume, $\mathbb{F}$ is a field of $char=0$ and $\mathbb{F}^\ast$ be the group of non-zero scalars. Let $G=\{(c,A)\in \mathbb{F}^\ast \times GL_n(\mathbb{F}) | AA^t=cI_{n\times n} \}$. Then $G$ is a ...
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Show subgroups of semidirect product are closed

This is Exercise 7.34 in Lie's book: Suppose $N$ and $H$ are Lie groups, and $\theta$ is a smooth action of $H$ on $N$ by automorphisms. Let $G=N\rtimes_\theta H$. Then I am trying to show that $\...
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For $N\unlhd G$ for $\mathfrak{B}$-group $G$ with $G/N$ a free $\mathfrak{B}$-group, show $\exists H\le G$ with $G=HN$ and $H\cap N=1$

This is Exercise 2.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". In the book, maps are evaluated from left to right. According to Approach0, the exercise is new to ...
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Is Symmetric group on 5 symbols is the semi-direct product?

Is the Symmetric group on 5 symbols is the semi-direct product of groups $A_5$ and $C_2$, i.e. $$S_5\cong A_5\rtimes C_2?$$ Here $A_5$ is considered as a normal subgroup. Please help.
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Role of semidirect product and intuition for $O(2)\simeq U(1)\rtimes \mathbb{Z}_2$?

I have a very basic understanding of some common groups, and I'm trying to get some intuition for this isomorphism. My thinking so far is that $O(2)$ is rotations and reflections in $\mathbb{R}_2$, ...
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$O(n)\cong SO(n)\rtimes O(1)$

I want to prove that $O(n)\cong SO(n)\rtimes O(1)$ as Lie groups. I have the following result: If $G,N,H$ are Lie groups, then $G\cong N\rtimes H$ iff there are Lie group homomorphisms $\phi:G\to H$ ...
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Characterization of permutable subgroups (direct product)

Edit: As pointed out by Mark, this question is obviously false when things are abelian. Please assume things are non-abelian. Let $G$ and $H$ be subgroups of a larger group, and define $GH = \{gh \,|\,...
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Naming the groups in a semidirect product

A group $G$ has the structure of an inner semidirect product when it can be reconstructed from two of its subgroups: one, often written $N \subset G$, is a normal subgroup, and the other one, $H \...
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Is the (outer) semidirect product, $N \rtimes H$, ever characterized as a group of bijections on a group containing $N$?

I'm an undergraduate who was introduced to semi-direct products of groups, $N \rtimes H$, relatively recently. At first, I found the multiplication rule for (outer) semi-direct products very annoying. ...
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1answer
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Identify subgroup of a semidirect product in GAP

I have created an external semidirect product of two groups $H$ and $K$ in GAP. Now, I want to identify $H$ as a subgroup of this semidirect product, but I am unable to identify $H$. How can I ...
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Understanding groups of order 12 as it appears in Dummit.

Question 1. Why the homomorphism $\varphi_1$ and $\varphi_2$ give rise to isomorphic semidirect product? From what i see, $\varphi_2=\varphi_1^2$ then $(V\rtimes_{\varphi_1} T)\simeq (V\rtimes_{\...
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How to show that semi-direct product is isomorphic to a direct product

On page 182 of Dummit and Foote Third Edition the following is stated during their classification of groups of order 30. I have worked through the entirety of the process, however I am confused at the ...
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Semi-direct product $C_p : C_3$

I want to construct the group $C_p:C_3$ in GAP, where $p$ is prime and $p\equiv 1 \pmod 3$. Originally, I come up with this ...
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How do I show that a group G is the semidirect product of two cyclic subgroups, one of order p^2 and the other of order p?

Let $p$ be an odd prime and $G$ be a group generated by elements $x,y$. Given $$x^{p^{2}}=1, x^{p} = y^{p}, yxy^{-1} = x^{p+1}, (yx^{-1})^{p} = 1,$$ show that $G$ is the semidirect product of a two ...
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How to prove that $G$ is semidirect product?

If $ N\triangleleft G $ and $ G / N $ is a free group, I want to show that $ G $ is a semi-direct product of $ N $ by $ G / N $, but I don't see how to prove it, I know if $ N $ is normal to $ G $, ...
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127 views

Elements and Conjugacy Classes of a group

Let $G=(C_{p_1} : C_{3}) \times(C_{p_2} : C_{3})$ where $p_1,p_2\equiv{1}\pmod{3}$. How many elements does the group $G$ have of each order? Furthermore, what is the total number of conjugacy classes?...
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Do we have to specify actions by many elements when defining some semidirect products

When studying about the semidirect product, $G=(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi} \mathbb{Z}_q$, I understood that, for some semidirect products by considering a minimal generating set $...
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Classification of groups of order $56$.

The particular problem that I am facing is that when $H$ is a normal subgroup of order $7$ and $K$ is a subgroup of order $8$ .Now, we can consider the map $\theta :K \to Aut(H)$ we see that $\theta(...
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Semi direct product characterization [duplicate]

Is there characterization of groups $G$ which can be written as the semi-direct of two other groups? One obvious necessary condition is that $G$ should have a normal subgroup. Is that the only ...
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Why is $A \rtimes B \simeq \mathbb Z \rtimes \mathbb Z/2\mathbb Z$

Hello everyone I have a hard time trying to resolve this problem if anyone could help it would be a lot appreciated. Let $$f_1\colon\mathbb R\rightarrow \mathbb R,\,f_1(x)=-x,\quad f_2\colon\mathbb R\...
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Subgroups of semidirect product of two abelian groups

Let $G= N \rtimes_{\varphi} Q$ be a semidirect product coming from a short exact sequence $$ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1 $$and assume that $N$ and $Q$ are finitely ...
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Is every subgroup of $\mathrm{Isom}(\mathbb{R}^n)$ a semidirect product $T \rtimes Q$?

It is well known that the group of isometries of Euclidean space $\mathrm{Isom}(\mathbb{R}^n)$ splits as the semidirect product $\mathbb{R}^n \rtimes O(n)$. However, is it also true that every ...
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Is there a way to check if a group extension is split or not using GAP?

I use the GAP command 'NormalSubgroups' to obtain the list of normal subgroups of a given group. Then I use 'FactorGroup' to construct the quotient of the given group by a normal subgroup. An example ...
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Classification of group extension. [duplicate]

Direct product, semidirect product and central extension are different types of group extensions. Is there any other type? If there is, can you please give an example with a small finite group?
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Showing that a Group Extension is Split

I have a group extension $1\rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ that I think is a split extension (so $G \approx N \rtimes Q$), but I'm having trouble showing this. Is there a ...
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inner semidirect and outer semidirect relationship

https://kconrad.math.uconn.edu/blurbs/grouptheory/group12.pdf. In the text above, the author explains how to find all groups of order 12. He does so by showing that a group $G$ (of order $12$) is ...
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$\mathbb Z_{15}$ inside the non-abelian group of order 75.

I know that any abelian group of order $75$ has a $\mathbb Z_{15}$ inside it.I was just wondering if there is a $\mathbb Z_{15}$ inside the non-abelian group of order $75$. If there is a normal ...
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1answer
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Cayley graphs obtained for different generating elements with the same orders in a finite group

When we consider the finite group $\mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime, $p>2$, the set with the pair of elements $\{(0,1), (1,0)\}$ can generate the group. Moreover, a set $\{(1,...
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Let $G=N\rtimes H$ and the conjugacy action of $H$ on $N$ divide $N$ in two orbits. Show $N$ is abelian (and more).

Given a finite group $G$, $N \trianglelefteq G$, and $H \leq G$, such that $G$ is the inner semidirect product of $N$ and $H$, and knowing that the conjugacy action of $H$ on $N$ divides $N$ in two ...
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Given $G = \mathbb{Z}_6 \times\mathbb{Z}_6$ with a particular operation, find $Z(G)$ and $G/Z(G)$

Given the group $G = \mathbb{Z}_6 \times\mathbb{Z}_6$ and the following operation: $$(a,b)\cdot (c,d)=(a+(-1)^bc,b+d)$$ find (or classify) $Z(G)$ and $G/Z(G)$. For the first part, I found that $(a,b) \...
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Hardness of the conjugacy search problem

I have come across the following problem, which is considered as a mathematically hard problem to solve (one way trapdoor) used for cryptography. Conjugacy search problem: Let $G$ be a non-abelian ...
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How to construct a topological group from a semidirect product of two subgroups

I'm trying to endow affine group $\mathrm{GA}(X)$, $X$ the given affine space, with canonical topology based on the semidirect product of two of its subgroups $T(X)$ and $\mathrm{GA}_a(X)$: $\mathrm{...
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45 views

Split short exact sequence contradiction

Am I doing it wrong or is this an exception to the split rules? There is a short exact sequence $$1 \longrightarrow \mathbb Z/_{2\mathbb Z} \hookrightarrow \mathbb Z/_{4\mathbb Z} \twoheadrightarrow \...
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Under what conditions is $V \rtimes_f \mathbb{k}$ isomorphic to $W \rtimes_g \mathbb{k}$?

Given a $\mathbb{k}$-vector space $V$ and an endomorphism $f$ of $V$, we can regard $V$ an an abelian Lie algebra and construct the semidirect product $V \rtimes_f \mathbb{k}$. It is given by the ...
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How is $\mathfrak{g} \ltimes_{\mathrm{ad}} \mathfrak{g}$ called?

Let $\mathfrak{g}$ be a Lie algebra. If $\mathfrak{h}$ is another Lie algebra and $\theta$ is a homomorphism of Lie algebras from $\mathfrak{h}$ to $\operatorname{Der}(\mathfrak{g})$, then we can form ...
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If $K$ is a normal subgroup of $G$ with $Z(K)=1$, then $G$ splits over $K$, iff $\operatorname{Aut}(K)$ splits over $\operatorname{Inn}(K)$

I want to prove the following proposition: Let $K$ be a normal subgroup of $G$ with $Z(K)=1$. Show that $G$ splits over $K$, if and only if $\operatorname{Aut}(K)$ splits over $\operatorname{Inn}(K)$. ...
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Prove that $G$ is the semidirect product $U\rtimes{D}$

Let be $k$ a field and $G$ the group of the triangular superior matrices in $GL_3(k)$. Prove that $G$ is the semidirect product $U\rtimes{D}$, where $U$ is the set of upper triangular matrices with $1$...

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