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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How many $3$-Sylow subgroups are there in $\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$ with $|\ker\rho| = 2$?

Let $G=\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$ with $|\ker\rho| = 2$. How many $3$-Sylow subgroups are there in $G$? I know that the number is $1$ or $7$, but I'm stuck.
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Automorphism group of elliptic curve in char 2

I'm trying to calculate the automorphism group of elliptic curve with $j$-invariant $0$ in a field $K$ of characteristic $2$. Let $ Y^2Z+b_3YZ^2=X^3$ the elliptic curve. The substitutions preserving ...
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$S_n \cong A_n \rtimes \{1,-1\} $

I have seen this question already a few times, but I still do not understand the actual answer. I have to prove the isomorphism between $S_n$ and the semidirect product of $\{1,-1\}$ and $A_n$. I am a ...
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Structure of non abelian finite p-groups

I am familiar with the concepts of direct products, semi-direct products, wreath products and central products of groups. After seeing the classification of finite $p$-groups upto order $p^4$,(Theory ...
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Quotient of $\mathbb R^n\rtimes O\left(n\right)$ by $\mathbb Z^n\rtimes D_8$

What is the coset space $\frac{\mathbb R^n\rtimes O\left(n\right)}{\mathbb Z^n\rtimes D_8}$ as a manifold? I saw a claim that it is the $n$-dimensional torus $\mathbb T^n=S^1\times\cdots\times S^1$? ...
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Split extension of finite group and Sylow subgroup by $p$-group

Let $p$ be a prime and let $A$ be an abelian normal $p$-subgroup of a finite group $G$. Hence, for any Sylow $p$-subgroup $H$ of $G$, it holds that $A$ is contained in $H$ and so $A$ is a normal ...
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Finding the relationship given by the generating elements of a non-abelian group (for a maximum case)

Let $G=(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$, where $p$ and $q$ are odd distinct primes. Let $G$ be generated by the elements $s=(g_1,e_q)$ and $t=(e_p,g_3)$, $g_1,e_p \in \...
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All non abelian groups of order $56$, when $\mathbb Z_7\triangleleft G$

I am calculating all non abelian groups of order $56$ when $\mathbb Z_7$ is a normal subgroup ( Given in the exercise $7$ of §5.5, Dummit Foote). I have searched for related posts on this site but ...
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Proof of $𝐶_𝐻(𝐺)∩𝐴=1$,$𝐻$ is a holomorph, $𝐴$ automorphism group

I was told that this was a simple proof, by definition of the centraliser and by definition of the intersection with the automorphism group, but I just don't see it. $A = \mathrm{Aut}(G)$ $H = G \...
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Dihedral group as a semidirect product?

It is known that the dihedral group $D_{2n}$ is isomorphic to the semidirect product $Z_n\rtimes Z_2$, where both $Z_n,Z_2$ are cyclic. My question is, for a semidirect prouct, the two subgroups ...
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Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2 $ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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If $G$ acts on $V$, how does $G^n\rtimes S_n$ act on $V^{\otimes n}$?

In a paper it was taken as obvious that if a finite group $G$ acts on a vector space $V$, then the semidirect product $G^n\rtimes S_n$ acts on $V^{\otimes n}$. I've tried to elaborate on how I think ...
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Representing a group as a quotient of a free group

Consider $G=F \rtimes T$, where $F=\mathbb{Z}_3 \times \mathbb{Z}_3$ and $T=\mathbb{Z}_5$. Let $\phi : \mathbb{Z}_5 \rightarrow Aut(\mathbb{Z}_3 \times \mathbb{Z}_3)$. It is said that any group is ...
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$M<K\rtimes H$ is a semidirect product?

Let $H,K$ be two finite groups, $K$ abelian, and let $M$ be a subgroup of $K\rtimes H$. Consider the projection $\pi:K\rtimes H \rightarrow H$ on the Second factor. Let us suppose that $\pi(M)=H$. ...
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Closed-form exponential and logarithmic map of Galilei group

I have a question regarding the logarithmic map $log: G\mapsto \mathfrak{g}$ and exponential map $exp: \mathfrak{g}\mapsto G$ between the Galilei group and its Lie algebra. The Galilei group of two ...
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Determining the homomorphism $\varphi: H \to \mathrm{Aut}(K)$ given a section $s: H \to G$

Per Wikipedia, a split extension is an extension $$1 \to K \overset{\beta}{\to} G \overset{\alpha}{\to} H \to 1$$ with a homomorphism $s: H \to G$ such that going from $H$ to $G$ by $s$ and then back ...
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Why does every $\varphi: K \to \mathrm{Out}(H)$ determine an unique extension of $H$ by $K$ when $Z(H) = 1$?

Every homomorphism $\varphi: K \to \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. Why is this true for groups $H$ with a trivial center? Even if we only consider split extensions, as ...
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2 cocycle and semidirect products

I am trying to solve the following question: Let (K, +) be an abelian group and let H be a group that acts on K by (group) automorphisms (k1, h1) • (k2, h2) = (k1 + h1 · k2 + ε(h1, h2), h1h2) Gε ...
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Constructing $\mathrm{SL}(2,5)\rtimes\mathbb{Z}_{11}^2$ in GAP

I am trying to construct the semidirect product $\mathrm{SL}(2,5)\rtimes\mathbb{Z}_{11}^2$ in GAP, where $\mathrm{SL}(2,5)$ is the subgroup $\left\langle\begin{pmatrix}4&1\\0&3\end{pmatrix},\...
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Is group extension unique?

Definition: Given groups N and H, a group is said to be an extension of H by N if there exists $N_{0}\vartriangleleft G$ such that $N_{0}\cong N$ and $G/N_{0}\cong H$ We know that a normal subgroup of ...
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When can we “multiply through” by the kernel of a homomorphism?

Let $G/H=K$. I think I am right in saying that if $K\cong K'\trianglelefteq G$ and is such that $K'H=G$ and $K'\cap H=1$ then we have $H\times K'=G$. If we relax the normality requirement of $K'$ to ...
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Coprime action and semidirect product

This is a article The Theory of Finite Groups. I'm reading lemma 8.2.1. http://web.math.ku.dk/~olsson/manus/GruFus/Kurzweil-Stellmacher_Theory%20of%20finite%20groups.pdf I don't why "In the ...
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64 views

Semidirect product and group action

I want to understand the following lemma: Let $G$ be a finite group satisfying $G = P \rtimes F$, where $P$ is a cyclic $p$-group for some prime $p$, $|F| > 1$ and $(p, |F|) = 1$. Then each ...
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Some clarifications required about the two extremes of general extensions (semi-direct products and central extensions)

This is a sequel to Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group? My professor made another remark that: Let's go ...
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Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group?

My professor made the following remark while teaching about group extensions: We want to classify finite groups in a manner similar to the fact that every positive integer is uniquely a product of ...
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Semidirect product of point stabiliser and regular normal subgroup.

Why is it that a permutation group $G$ on $\Omega$ with regular normal subgroup $K$ is a split extension (internal semidirect product) of $K$ and the point stabiliser $G_\alpha$ for some $\alpha\in\...
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Why is trivial intersection of groups $N$ and $H$ not required in the definition of outer semi-direct products?

In the case of outer (synthetic) semi-direct products, we take any two groups $N$ and $H$ and a group homomorphism $\varphi: H \to \mathrm{Aut}(N)$ and effectively "synthesize" a new group named $(N \...
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Clarification regarding condition of semidirect product of cyclic groups

Wikipedia says: More generally, a semidirect product of any two cyclic groups $C_m$ with generator $a$ and $C_n$ with generator $b$ is given by one extra relation, $aba^{−1} = b^k$, with $k$ and $n$ ...
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Wreath product, semi-direct product, and partitions

Any help with the question which follows will be greatly appreciated. I'm working through Dixon and Mortimer's Permutation Groups and have a question regarding a particular semi-direct product ...
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Subgroups of index $p^2$ of $(\mathbb{Z}/p^2\mathbb{Z})^\times\ltimes \mathbb{Z}/p\mathbb{Z}$

I want to prove that the group $(\mathbb{Z}/p^2\mathbb{Z})^\times\ltimes \mathbb{Z}/p\mathbb{Z}$ has a unique subgroup of index $p^2$. I know that $(\mathbb{Z}/p^2\mathbb{Z})^\times$ is cyclic of ...
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When the direct and semi-direct products are isomorphic.

I am not asking what direct/semi-direct products are. Suppose $H$ and $K$ are any two groups, and let $\varphi:K\to\text{Aut}(H)$ be a homomorphism, and consider the semi-direct product $H\rtimes K$ ...
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Centralizer of symmetric group

Let, an element of symmetric group $S_N$ is given by $g=(1)^{N_1}(2)^{N_2}....(s)^{N_s}.$ Here $N_n$ denotes the number of cycles of length $n$. Its known that the centralizer of this element is ...
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Computing Lie algebra of semidirect product

I have the following problem. Given the following homomorphism $\phi:\mathbb{R}\rightarrow$Aut($\mathbb{R}^2$), $\phi(t)$= $\left(\begin{matrix} e^{kt} & 0\\ 0&e^{-kt}\\ \end{matrix}\...
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Generators of $C_3\rtimes C_2$

Can I write elements of $G=C_3\rtimes C_2$ as $$\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\}?$$ Then, what are the generators of $G$? $(0,1)$ and $(1,0)$? I've learned that the multiplication of semi-...
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Elements of $C_3\rtimes C_2$ not $S_3$ or $D_3$

Can we represent an element of $G=C_3\rtimes C_2$ as $(a,b)$ like we do in the direct product? Because when I draw a Cayley diagram of $G$, I don't know how to label each node and arrow without the ...
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On the number of subgroups of semidirect product

Let $G=H\rtimes K$ be the semidirect product of the normal subgroup $H$ and the subgroup $K$. Let $D$ be a subgroup of $G$ such that $K\le D$. Does $D= (D\cap H)\rtimes K$?. I believe that is true, ...
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Proving that the $D_n$ is isomorphic to the semi-product group formed by $C_n$ and $C_2$.

Let $D_n$ denote the dihedral group for a regular $n$-gon with $n \geq 3$. Show that $D_n$ has a semidirect product structure, $$D_n \cong C_n \rtimes_\varphi C_2.$$ What is $\varphi:C_2 \to\...
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A group of order $340$ is isomorphic to $H\rtimes K$, with $K$ being a $2$-Sylow subgroup

The first part of the exercise asked us to show that there's a normal, cyclic subgroup $H$ of order $85$. That's pretty easy using Sylow's third theorem, and generating a subgroup with elements of ...
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semidirect product between subgroup of general linear group and vector space in GAP

I am currently working on trying to get a solvable doubly transitive permutation group using GAP. So, I am trying to create the semidirect product of a subgroup of a general linear group and a vector ...
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160 views

Semidirect product action and its geometry

I'm going by the maxim Groups, like men, are known by their actions This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does ...
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33 views

Inner semidirect product as outer semidirect

Let $G$ be an (ambient) group which splits into an (inner) semidirect product $G_1\rtimes_\varphi G_2$ of two subgroups (satisfying other conditions). We can define $\varphi:G_2\to\mathrm{Aut}(G_1),\...
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The natural action of $SL_2(\Bbb R)$ on $\Bbb R^2$

Suppose we have special linear group $SL_2(R)$, we can construct a semi-direct product $\Bbb R^2 \rtimes SL_2(\Bbb R)$ through the natural action of $SL_2(\Bbb R)$ on $\Bbb R^2$. What is the ...
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Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product.

Question: Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product. (This is a question from Rotman's Advanced Modern Algebra which I am ...
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84 views

Groups which cannot be written as semidirect products

I am reading Dummit & Foote, Abstract Algebra, 3e, p.103ff. We know that the first part of Jordan-Hölder program, the classification of finite simple groups, are finished. But it is not written ...
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96 views

Number of subgroups of semidirect product of two elementary abelian subgroups

It is well known that a subgroup of the semidirect product $H\rtimes K$ is not in general semidirect product of two subgroups $H'\le H$ and $K'\le K$ but always exist some subgroups of $H\rtimes K$ on ...
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Group action by a semi-direct product.

Let $G$ be a group and $H,N$ subgroups, of which $N$ is a normal subgroup. Suppose that $G= H \ltimes N$ and that $H \cap N = 0$. Is any action of $G$ on a set $X$ equivalent to an action of $N$ on ...
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103 views

Presentation $\langle x,y \mid x^3=y^3=(xy)^3=1\rangle\cong\langle t\rangle\ltimes A$

Hi: This question has already been answered here: Show $\langle x,y|x^3=y^3=(xy)^3=1\rangle$ is isomophic to $A\rtimes\langle t\rangle$, where $t^3=1$ and $A=\langle a\rangle\times\langle b\rangle$. ...
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Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$

Let $A$ and $B$ be matrices of finite order with integer coefficients. Let $n\in\mathbb{N}$ and let $G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^n$ be the semidirect product, where the action is $\varphi(n)\...
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1answer
69 views

Existence of elements in $\text{Gal}(\mathbb{Q}(\sqrt[5]{2}, \xi)/\mathbb{Q})$ with $\xi = e^{\frac{2\pi i}{5}}$

Consider $p(x) = x^5-2 \in \mathbb{Q}[x]$. I have proven that the splitting field of $p(x)$ is $F = \mathbb{Q}(\sqrt[5]{2}, \xi)$, where $\xi = e^{\frac{2\pi i}{5}}$. After doing so, I have proven ...
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46 views

Showing that two elements generate $\mathbb Z[\frac13] \rtimes \mathbb Z$.

Let $\mathbb Z$ act on the additive group $\mathbb Z\left[\frac13\right] = \{a/3^k : a \in \mathbb Z, k \ge 0\}$ by $\varphi_n(r) = 3^n r$ for $n \in \mathbb Z$ and $r \in \mathbb Z\left[\frac13\right]...

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