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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Classification of groups of order $20$ [duplicate]

I am trying to classify the non-abelian group of order $20$. We know that $K$ is a subgroup of order $5$ is normal. $H$ is a subgroup of order $4$. Case $1:$ let $H \cong \mathbb{Z_2} \times \mathbb{...
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Defining semidirect product and presentation when one of the groups is a product of cyclic groups

I'm trying to classify the groups of a certain order and have shown that a group $G$ with that order can be expressed as the semidirect product of a normal subgroup $N$ $\cong$ $C_n$ $\times$ $C_m$ ...
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If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can the fixed points form a non-normal subgroup?

Suppose that we have a finitely generated and residually finite group $G = K\rtimes\mathbb{Z}$, but $K$ is not finitely generated. Let $T$ be a finite subset of $K$ such that $\langle (0,1), (k,0)\mid ...
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Classification of groups of order $18$

I was just going through the first classification- $$|G|=18=3^2\times 2$$ Then $G$ has a subgroup of order $9$(normal, say $K$) and a subgroup of order $2$ (say $H$). I want someone to help me with ...
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How do we find the homorphism from $\mathbb{Z_2} \to{\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ [closed]

How do we find the homorphism from $\mathbb{Z_2} \to {\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})?$ I know that ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ is isomorphic to $GL_2(\mathbb{Z_3})$. We ...
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Perfect semi direct products

Let $ \pi,V $ be a representation of a perfect group $ G $. I'm interested in sufficient conditions for a semi direct product like $ V \rtimes_\pi G $ to be perfect. Requiring that $ \pi $ is faithful ...
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$p,q$ primes, $p\mid q-1$. Weaker assumption in the proof of the existence of non-trivial $C_p\ltimes C_q$?

Motivated by the fact that the non-existence of non-trivial $C_p\ltimes C_q$, for $p\nmid q-1$, can be proven without any piece of information on the structure of $\operatorname{Aut}(C_q)$, not even ...
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Generalization of Semidirect Product (Finite Groups with non-trivial minimal subgroup)

Suppose any two non-trivial (non-singleton) subgroups of a group $G$ have a non-trivial intersection. $|G|$ is necessarily a prime power, because by Cauchy's theorem, for any prime $q$ dividing $|G|$, ...
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Groups of the form $(\Bbb Z_{13} \times \Bbb Z_7)\rtimes \Bbb Z_3$

Consider the semi-direct product: $(\Bbb Z_{13} \times \Bbb Z_7)\rtimes \Bbb Z_3$ To construct a group $G$, we need homomorphisms $\theta$: $\Bbb Z_3 \rightarrow \text{Aut}(\Bbb Z_7)$ and $\theta_2$: $...
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Show $S_3$ is the semi-direct product of $A_3$ by $H=\{(1),(12)\}$

Show $S_3$ is the semi-direct product of $A_3$ by $H=\{(1),(12)\}$ How would you prove such a question? Showing $H \cap N = \{id\}$ is fine, where $N=A_3$, but how would I show that $A_3H=G$ for semi-...
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If $K\rtimes \mathbb{Z}$ is a finitely generated and resdiually finite group but $K$ isn't, can the following abelianization all be finite?

I am looking for a residually finite semidirect product with the following properties. This is related to this question: If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, can ...
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Classify groups of squarefree order $pqr$.

Given distinct primes $p,q,r$, how many groups of order $n=pqr$ do we have given that: I) $q, r = 1\pmod p$ and $r = 1 \pmod q$ II) $q,r=1\pmod p$ but $r \neq 1 \pmod q$ III) $q = 1\pmod p$ and $r = 1\...
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Weyl group of $C_2$ in $C_2 \ltimes U(n)$

Let $C_2$ denote the cyclic group of order $2$ and let $C_2 \ltimes U(n)$ denote the semi-direct product of $C_2$ with the unitary group, where $C_2$ acts on $U(n)$ by complex conjugation. I want to ...
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Is this semi-direct product residually finite?

Consider the group $G=K\rtimes \mathbb{Z}$ defined as follows: The subgroup $K$ is generated by elements $x_i,y_k$ with $i,k \in {\mathbb Z}$ and $k > 0$, and it has defining relations \begin{...
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Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$?

Let $G= A \rtimes B$ where $ A,B \leq G$. ( where $ \rtimes $ denotes semi-direct product) Question- Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$ ? If yes, ...
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Center of split extension of groups of co prime orders [closed]

Let $G = A \rtimes B,$ where $A$ is a finite $p$-group and $B$ is a finite $p'$ group. When can we say that $Z(G) \subseteq A?$ For example the center of the group $C_3 \rtimes C_4$ is $C_2$ and hence ...
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Is there any factorization of Leibniz algebras?

A semigroup S is factorisable if there are subsemigroups A and B such that S = AB. In the case of Leibniz algebras, can we say that a Leibniz algebra is a direct sum of two subalgebras? Any reference, ...
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Is $\operatorname{Hol}(D_4)$ isomorphic to a familiar group?

We define the holomorph of a group, $\operatorname{Hol}(G)$, as its semidirect product $G\rtimes _f\operatorname{Aut}(G)$. As it happens (as is shown here), $D_4\approx\operatorname{Aut}(D_4)$, and we ...
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In $\operatorname{Hol}(\Bbb{Z}_{10})$, is every mapping from $G\rightarrow\operatorname{Aut}(G)$ just an identity mapping?

Consider $f: G\rightarrow\operatorname{Aut}(G)$ where $G=\Bbb{Z}_{10}$. Since this is equivalent to considering $G\rtimes_f G$, we know that $f$ is defined as $f_a(b)=aba^{-1}$. There are four ...
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If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

I recently answered the following question: If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can there be fixed points? I have a related question: Is ...
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Is this a valid usage of the semidirect product?

I've been trying to wrap my head around the semidirect product of two groups, and I think I'm starting to get it, but I wanted to double check if this is true or not. If $N$ and $H$ are subgroups of $...
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If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can there be fixed points?

Suppose that we have a group $G = K\rtimes\mathbb{Z}$, where $G$ is finitely generated, but $K$ is not finitely generated, and let $\phi(1)$ be the automorphism of $K$ corresponds to $1_\mathbb{Z} \in ...
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What is a chief series of a group $H=(\mathbb{Q}_8 \rtimes C_3 ) \times A_5$?

The definition of Chief series is given in a link. Question: Consider the group G= $\mathbb{Q}_8 \rtimes C_3 $ (where the action is non trivial), its chief series is $$ e \triangleleft Z_2 \...
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A basic question regarding semidirect product of finite groups.

I have a basic question regarding the semidirect product $C_m\rtimes C_k$ of two finite cyclic groups. Does the semidirect product $C_m\rtimes C_k$ represents a specific group or rather a family of ...
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Strategy for classifying some groups of order $pqr$ - recognizing direct factors

I've been reviewing some of my notes from an abstract algebra class that I took and have been thinking about/redoing some of the examples we did classifying groups of smallish order $pqr$. In ...
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Defining relation on semidirect product of groups

$\newcommand{\aut}{\operatorname{Aut}}$ I wonder how we can extract defining relation on semidirect product of groups. Consider a group of order $12$. Consider two cyclic groups $C_3 = \langle a\...
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If $H$ and $K$ are non-isomorphic but of same order, can there be $\varphi_1, \varphi_2$ such that $N\rtimes_{\varphi_1}H\cong N\rtimes_{\varphi_2}K$?

Let $N$, $H$ and $K$ be finite groups, such that $|H| = |K|$ and $H$ and $K$ being non-isomorphic. Can there exist $\varphi_1$ and $\varphi_2$ such that $ N \rtimes_{\varphi_1} H \cong N \rtimes_{\...
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3 answers
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Semidirect product and direct product

Let $A$ be an abelian group and $G$ a group and let $\alpha:G\rightarrow{\rm Aut}(A)$. I want to show that the semidirect product $A\rtimes _{\alpha }G$ is isomorphic to the direct product $A\times G$ ...
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Semidirect product of free abelian groups

Consider the semidirect product $$ G=\mathbb{Z}^n \rtimes \mathbb{Z}^m $$ Is $ G $ always virtually abelian? Is it the case that the abelianization of $ G $ is $ \mathbb{Z}^{n+m} $ if and only if $ G $...
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Finding the inverse of a group isomorphism linked to a short exact sequence

Exercise: Construct an explicit inverse of $\phi$ in the following theorem Theorem: Given a short exact sequence with a right split (this is, a $v: K \to G$ such that $\epsilon v = 1$), which ...
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If the Sylow-$2$-subgroup of a group $G$ is cyclic then can it surject $\mathbb{Z}_2\oplus\mathbb{Z}_2$

Suppose $G$ is a group such that Sylow-$2$-subgroup is cyclic. Then can it surject into $\mathbb{Z}_2\oplus\mathbb{Z}_2$? If $G$ is an abelian group and Sylow-2-subgroup is Cyclic, then it can not, by ...
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Prove this isomorphism about semidirect product $((\oplus_1^k \mathbb{Z}/2) \oplus \mathbb{Z}/n) \rtimes_{\varphi} \mathbb{Z}/2$.

Edited: Considering the comments it turns out that my guess was true, and we have this isomorphism: $$((\oplus_1^k \mathbb{Z}/2) \oplus \mathbb{Z}/n) \rtimes_{\varphi} \mathbb{Z}/2 \equiv (\oplus_1^k \...
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6 votes
2 answers
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Characterization of the isomorphic semidirect products

Let $A$ and $G$ be two finite abelian groups and let $\alpha$, $\beta:G\rightarrow{\rm Aut}(A)$. Suppose that $\alpha (G)$ and $\beta (G)$ are conjugate subgroups of ${\rm Aut}(A)$. Are the semidirect ...
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Showing, for $G=\langle\delta\rangle\ltimes(A\times A)$ and abelian $A$, that $Z(G)=G'\cong A$.

This is the second part of Exercise 5.2.2(a) of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. Here is the previous part: Show $G=...
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A question about equivariance to 3D transformations using semi-direct and direct products.

in the paper: https://arxiv.org/pdf/2010.02449.pdf the author consider some classes of functions that are invariant to the action of the group $G= \mathbb{R}^3 \rtimes SO(3) \times S_n$ on $\mathbb{R}^...
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Showing the existence of an automorphism between two semidirect products

Let $H$ and $K$ be two groups, and let $\phi$ and $\phi’$ be two group homomorphisms from $H$ to $Aut(K)$. Then it can shown that if there’s an automorphism $\sigma$ in $H$ such that $\phi’=\phi \circ ...
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Find all semidirect products $\mathbb{Z}_{8} \rtimes \mathbb{Z}_{4}$ up to isomorphism

I am trying to solve this: Find all semidirect products $\mathbb{Z}_{8} \rtimes \mathbb{Z}_{4}$ up to isomorphism First I tried to find all homomorphisms $\varphi:\mathbb{Z}_{4} \to{\rm Aut}(\mathbb{...
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Obtaining splitting of quotient map from semi-direct product

Suppose $N \lhd G$. Given isomorphism $i: N \rtimes_\pi G/N \to G$ and action $\pi: G/N \times N \to N$, show how to obtain a splitting $\varphi: G/N \to G$ of the quotient map $G \to G/N$. Here, we ...
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Group of order $pq$

I am struggling with this exercise: Let $G$ be a group of order $pq$, where $p > q$ and $p, q$ are primes.Prove that if $q|p−1$, then either $G \cong \mathbb{Z}_{pq}$ or $G \cong \mathbb{Z}_{p} \...
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Show $G=\langle\delta\rangle\ltimes D$ is nilpotent of class $2$.

This is part of Exercise 5.2.2(a) of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. It is marked as being referred to later on in ...
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Show that $G= \langle x,y\mid x^2,xyx^{-1}y=e \rangle$ is a semidirect product of two of its subgroups

Show that $G= \langle x,y\mid x^2,xyx^{-1}y=e \rangle$ is a semidirect product of two of its subgroups my first attempt was to use the theorem: $G\cong H\rtimes K$ iff $H=$ normal $H\cap K=\{e\}$ and ...
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If a short exact sequence splits we have a semidirect product

There are a lot of similar questions concerning this fact, and I looked through them all but the answers tend to take larger steps and I'm having a hard time following them. Suppose that $1\to L \...
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Understanding the definition of group split extensions

I’m reading about group extensions and I’m finding hard to understand how the following definition (from Wikipedia): A split extension is an extension $1\to K\to G\to H\to 1$ with a homomorphism $s\...
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1 vote
1 answer
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If $H \unlhd G$ then is $H \unlhd G \rtimes K$

Suppose we have a semidirect product $G \rtimes_{\varphi} K$ and $H \unlhd G$, then is it necessarily true that $H \unlhd G \rtimes_{\varphi} K$? We know that $G \unlhd G \rtimes_{\varphi} K$ but this ...
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Which groups are a non-trivial semidirect product of abelian groups?

Fact 1 (see this MSE post): The semidirect product $G \rtimes H$ is abelian iff $G$ is abelian, $H$ is abelian, and the semidirect product is trivial (and thus is just a product.) So we may ask, ...
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Show that $G=S_3S_5\rtimes S_2$ where $|G|=30$ and $S_p$ is a $p$-Sylow subgroup

Let $G$ be a group of order $30$ and let $S_p<G$ a $p$-Sylow subgroup for $p=2,3,5$. Show that $G=S_3S_5\rtimes S_2$ (interior semi direct product). I managed to show that $S_3S_5\cap S_2=\{e\}$ ...
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Computing time complexity and computational cost in non-abelian group based computation

I came across the following questions regarding the time complexity and the computational cost. They are related to computations in non-abelian groups. Suppose $H \rtimes_{\phi} K$ is a semidirect ...
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$C_4\rtimes C_2$ isn't isomorphic to $C_4\times C_2$ [duplicate]

Explain why $C_4\rtimes C_2$ isn't isomorphic to $C_4\times C_2$ My first thought was to say that since $C_4\rtimes C_2 \simeq C_4$ then if $C_4\rtimes C_2\simeq C_4\times C_2 $ we would get $C_4\...
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Information about $P'$ in $P \rtimes Q$

Let $p$ and $q$ be distinct primes. Suppose we have a group $G = P \rtimes Q,$ where $P$ is a non-abelian $p$-group and $Q$ is an abelian $q$-group such that it is a subgroup of ${\rm Aut}(P).$ Then ...
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2 votes
2 answers
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Generating sets of semi-direct products with $\mathbb{Z}_2$

Suppose a group $G$ splits as a semidirect product $N\rtimes\mathbb{Z}_2$, and let $\phi:G\to\mathbb{Z}_2$ the the associated quotient map. If I have a subset of elements $\{g_1,\dots,g_n,h\}$ of $G$ ...
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