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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Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $ U $ on the diagonal. Find a Sylow-p-...
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Subgroup of semidirect product

Let $G$ be a semidirect product of a normal subgroup $A$ with a subgroup $B$ and Let $H$ be a subgroup of $G$ such that $H\cap A$ is trivial. Is it true that $H$ is contained in a conjugate of $B$ ? ...
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External Semidirect product and isomorphism

Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism. Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ ...
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problem about inner semidirect product

Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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subgroup of a semidirect product

I'm really lost with this problem and I really need your help: Let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$, and let $H\leq G$ with finite index in G. I have to prove that there is a subgroup $U$ of $\...
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Direct Product and Semi-direct Product between $S_{3}$ and $Z_{2}$, and Related Problem

I am trying to figure out the following question: 0) Is the direct product $S_{3}\times Z_{2}$ isomorphic or non-isomorphic to the semi-direct product $S_{3}\rtimes_{\phi}Z_{2}$ where $\phi:Z_{2}\...
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Subgroups of the Semi-Direct Product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$

I want to list all the subgroups of the semi-direct product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$, under the homomorphism $\theta: (\mathbb{Z}/7\mathbb{Z})^{\times} \...
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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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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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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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Is this intuition for the semidirect product of groups correct?

My abstract algebra class introduced me to direct products, not semidirect products. I became interested in semidirect products when confronted with the following homework problem: Define the ...
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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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Do homomorphisms $H \to \operatorname{Aut}(K)$ that coincide at the level of $\operatorname{Out}(K)$ induce isomorphic semidirect products?

I recently realized that I don't know of any group that is a nontrivial semidirect product of some symmetric group $S_n$ and another group ($S_n$ being the normal subgroup), except when $n=6$. (For ...
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Find a non-trivial outer automorphism of $A_n$?

The symmetric group $S_n$ can be decomposed as a semi-direct product of the alternating group $A_n$ and a subgroup of $S_n$ of order 2. Use this fact to find a non-trivial outer automorphism of $...
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If $G$ is cyclic of order $2$ show $|H^1(G,\mathbb{Z})|=2$.

Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has ...
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Intuition about the semidirect product of groups

If we have two groups $G,H$ the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product $G\times H$ into a group it is certainly by ...
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Why isn't $1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$ a split extension? [duplicate]

Let $\mathbb{Z}_n$ be the cyclic group of $n$ elements and $1$ the trivial group. I'm looking at the following extension: $$1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$$ Because every ...
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Fundamental group of Mapping Torus $M_h$: how to prove that the action is really $h_*$?

Let us work with the following setting: let $h$ be an automorphism (assume base point preserving) of a genus $g$ surface ($g>0$) to himself. $h \colon (\Sigma^g,\ast) \to (\Sigma^g,\ast)$ and ...
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Cayley table for semidirect product $\mathbb{Z}_3 \rtimes _\alpha \mathbb{Z}_2$?

Let $\alpha : \mathbb{Z}_2 \rightarrow Aut(\mathbb{Z}_3) \cong \mathbb{Z}_2$ be the homomorphism given by $\alpha_{\bar{1}}(\bar{1}) = \bar{2} \in \mathbb{Z}_3$. Write down the Cayley table for $G=\...
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Semi-direct product Lie algebra

Given Lie algebras $S$ and $I$ and a Lie homomorphism $\theta:S\to Der I$, we have the semidirect product to be the space $S\oplus I$ with operation $$ (s_{1},x_{1})(s_{2}x_{2}):=([s_{1},s_{2}],[x_{1},...
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Number of groups of order $n$ being a semidirect product of non-trivial groups?

It can be extremely difficult to determine the number of groups of order $n$ (upto isomorphism). Suppose a number $n$ is given and for every proper divisor $d|n$, we know the number of groups of ...
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Quotient group of a semidirect product.

Let $G=AB$ be a semidirect product of groups $A$ and $B$ with $B\lhd G$ and $A\cap B=\{e\}$. Let $N=[A,B]$ be the commutator subgroup where any commutator $[a,b]=aba^{-1}b^{-1}$. 1.) Show that $N$ is ...
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How can I check whether a given finite group is a semidirect product of proper subgroups?

Suppose, a finite group $G$ is given. I want to check whether there is a proper normal subgroup $N$ of $G$ and a subgroup $H$ of $G$, such that $G$ is the semidirect product of the groups $N$ and $H$...
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Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi H \;...
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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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Inverse element in semidirect product

If K and Q are both groups and $h:Q\rightarrow \text{Aut}(K) $ is a homomorphism then the group operation for the semidirect product $K\rtimes_hQ$ is: $$(k_1,q_1)*(k_2,q_2)=(k_1h(q_1)(k_2),q_1q_2)$$ ...
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Multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$

I am trying to compute the multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$. My effort: I understand that $S_1$ is the trivial group consisting only the unit element i.e. $e$. $e$ will ...
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Automorphism group of planar graphs

I am going through Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. It has shown that the automorphism group of a planar graph $G$ is as follows. $$ \text{Aut} \left(G\...
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Are subgroups of automorphism groups of trees direct product of symmetric, cyclic and dihedral groups?

My question is triggered by my confusion with the notation $\Psi$ in Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. The notation $\Psi$ was first used expressing ...
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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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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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How to determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$?

How can I determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$ ? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of ...
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How to prove $\sum^t_i S_{N_i} \wr D_{m_i}$ non-Abelian?

How do I prove that $\sum^t_i S_{N_i} \wr D_{m_i}$ is a non-Abelian group? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
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Semidirect product of two groups defined in terms of a homomorphism

I am going through From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups by Bacon et al. I am having trouble to understand the ...
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Is semidirect product unique?

This is about semi direct product on Dummit and Foote algebra text book. Why is this statement true? Theorem 12. Suppose $G$ is a group with subgroups $H$ and $K$ such that $H\...
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How to find the images of the external semidirect product?

Once we have two groups, let's say $H$ and $G$, it's fairly simple to find all split extensions of $G$ by $H$ by looking at the automorphism group $\text{Aut}(H)$ and homomorphisms $\tau: G \...
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Coxeter presentation of Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$.

I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation $$\langle s_{\text{1}},\ldots,s_n\mid s_{\text{1}}^{\text{2}}=s_i^2=1, (s_1s_2)^4=(s_is_{i+1})^3=(...
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Automorphisms of the affine semilinear group $A\Gamma L(1,2^{n})$

In this question, it is mentionned that the group of automorphisms of the semilinear group $A\Gamma L(1,2^{n})$ is the group itself. Do you have a short proof of this fact?
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General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
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Interpreting $S_{N} \wr D_{m}$

I am trying to interpret $S_{N} \wr D_{m}$ in the light of the interpretation of $\mathbb{Z}^n_2 \wr \mathbb{Z}_2$ in this paper. So, according to the definition, $$\mathbb{Z}^n_2 \wr \mathbb{Z}_2 =...
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Application of the semi-direct product

how would you prove that $Z/30Z$ x $Z/7Z \simeq Z/210Z$ using the semi direct product? I know that if |G|=pq, (p and q primes) there exist an homomorphism $\phi: C_p --> Aut C_q $ such that G $\...
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Semidirect product: general automorphism always results in a conjugation

When $G$ is a group, $N$ is a normal subgroup of $G$ and $H$ is another subgroup of $G$ where $ N \cap H = \{1\} $, the normality of $N$ suggests that we can write, for $n_1, n_2 \in N$ and $h_1, h_2 \...
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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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About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$ [closed]

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? ...
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Progressed A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest

I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery: Let $ H $ be a group and $ K = \langle x\rangle $ be a cyclic group (...
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When is a group a semi-direct product with its normal subgroup?

Let $G$ be an infinite non-abelian group. If we have a normal subgroup $N$ of $G$, then can we always construct the subgroup $H$ such that $G$ is a semidirect product of $N$ and $H$?
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Symmetric group isomorphic to semidirect product of Alternating group and Z/2Z

I'm having a hard time understanding why $A_n \rtimes \mathbb{Z}_2 \cong S_n$. I understand that $A_n$ is normal in $S_n$. But that's about it. What would the $\alpha$: $\mathbb{Z}$$_2$$\...
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Questions related to semidirect-product of Klein four group?

I have four questions related to Klein four group. and I know the answer two of them. ( the first two) and I want to know answer Does $V_4$ has an automorphism of order 6? What are the orders of ...
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Borel subgroup is a semidirect product of the subgroup of all unitriangular matrices and the maximal split torus

Suppose $G = GL(n,q)$ and $B\leqslant G$ is the Borel subgroup, $N \leqslant G$ is the subgroup of all monomial matrices and $T=B\cap N$ is a maximal split torus. I am trying to understand why $B=U \...
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Constructing semidirect product out of finite fields and Galois groups and the permutation groups they induce

Let $r$ be a prime and let $K$ be a finite field of order $2^r$. Let $A$ denote the addtive group of $K$, let $M$ denote the multiplicative group of $K$ and let $H$ denote the Galois group of $K$ over ...