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 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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40 views

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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33 views

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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57 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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46 views

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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37 views

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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67 views

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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54 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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55 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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72 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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94 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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41 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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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 ...
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40 views

SemiDirect Products-Recognizing when they are isomorphic or not

I know there are methods of showing when $H\rtimes_{\Psi_{1}}K$$\cong$$H\rtimes_{\Psi_{2}} K$. However, what about semidirect products in which the H's differ. Is it ever the case where $|H_{1}|=|...
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36 views

Is semi-direct product represent all group (up to isom) of order 6?

We have $|G|=6$ and by sylow thm we have $n_{3}=1\ n_{2}=1\ or\ 3$ this implies $\exists H \in Syl_{3}(G)$ and $H\lhd G$. Let $K \in Syl_{2}(G)$, then $H\cap K = {1}$ and $HK=G$ and $G\cong K\times_{\...
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19 views

Computing the characters of $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $

Let a group be $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $ where $t \in \mathbb{Z} \text{ and } t \ge 1; N_i, m_i \in \mathbb{Z} \text{ and } N_i, m_i \ge 1$; and $S_{N_i}$ is a symmetric group over $N_i$ ...
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151 views

Why $W(B_\ell) \simeq (\Bbb Z/2\Bbb Z)^\ell \rtimes S_\ell$?

Let $V:=\Bbb R^\ell $, $\ell \geq 2$ with an orthonormal basis $\{e_i\}_{i=1}^\ell $. The set $$\Phi = \{ \pm e_i \pm e_j \mid 1 \le i\neq j \le \ell\} \cup \{ \pm e_i \mid 1 \le i \le \ell\} $$ is ...
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94 views

Commutative diagram of semidirect products

My question is partly motivated by trying to solve this one. Let $E$ be the semidirect product of groups $G$ and $H$. Then, we have an exact sequence: $$G \hookrightarrow^\iota G \rtimes_\phi H \...
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176 views

semidirect product of isometry group

I am doing exercise about semidirect product. Here is the question: Prove that the isometry group of Euclidean space $R^n$ is $O(n)\rtimes R^n$. I was stucking. Any ideas?
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398 views

Order of elements in Semidirect product of groups

Let $G$ and $H$ be two finite groups, where $H$ acts on $G$ trivially, so that $\eta_h(g)=g$ for all $g\in G$ and $h\in H$, and $G$ acts on $H$ by conjugation. We want to construct the semi direct ...
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157 views

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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102 views

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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82 views

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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440 views

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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43 views

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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148 views

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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19 views

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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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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141 views

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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69 views

Show a group is a semi direct product

Consider a set $G$ of $n \times n$ matrices with entries $\{0,1,-1\}$ that have exactly one non zero entry in each row a column. Show $G$ is a group and that $G$ is the semi direct product of the ...
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62 views

Prove the following two groups are isomorphic

I'm looking for an answer to the following problem, any responses would be greatly appreciated as I think it's quite complicated!. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector ...
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44 views

Semidirect product.

I have a problem with representation of this : $(D_n \times D_n) \rtimes \mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $D_n \times D_n$ by exchanging the two components. $D_n = \langle x, \ y \ | \ x^n ...
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65 views

Show that the group of permutations of $\{1,2,3,4\}$ is equal to the product of it's subgroups…

Show that the group of permutations of $\{1,2,3,4\}$ $$\sigma_4$$ is equal to the product of it's subgroups $$C_2\times C_2 $$ and$$D_6=(x^3=y^2=1, yx=x^2y)$$ I'm not sure whether to just multiply ...
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507 views

Automorphism group of a semi-direct product

I'm trying to construct the semi-direct product $(\mathbb{Z}_7 \rtimes \mathbb{Z}_3) \rtimes \mathbb{Z}_2$. Constructing the first factor in parentheses is not difficult. But when it comes to ...
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Non abelian group of order 1575

Construct a non abelian group of order $1575$. I am sure I am to use semi direct product.Give me some idea/hint to start.
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Finding the normal subgroups in a semidirect product

Let the group $A=\cdots\times\mathbb{Z}_{-1}\times\mathbb{Z}_{0}\times\mathbb{Z}_{1}\times\cdots$ with $\mathbb{Z}_{i}=\left\langle a_{i}\right\rangle $ and $\alpha:a_{i}\rightarrow a_{i+1}$ an ...
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ses and outer semidirect products of groups

Let $\phi : H \to Aut(N)$ be an action ($N,H$ are groups), and let $G = H \rtimes_\phi N$ be the semidirect product ($N$ is normal in $G$). I know that when $\phi$ is the action by conjugation, there ...
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173 views

Describe up to isomorphism the semidirect product

I have the following problem: I have to describe up to isomorphism the semidirect product $C_6 \rtimes C_2$, where $C_6$ denotes cyclic group of order six. I think I have to use external semidirect ...
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102 views

$ K \rtimes_{\phi_1} C \cong K \rtimes_{\phi_2} C$ when $\phi_1(C), \phi_2(C)$ are conjugated and $C$ is a product of two cyclic groups

I know the following result: Let $C$ be a finite cyclic group, $K$ a finite group such that there exist homomorphisms $\phi_1,\phi_2$ $\phi_i:C \to Aut(K) $ such that $\phi_1(C), \phi_2(C)$ are ...
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27 views

Computing the tangent map of the conjugation in a semidirect product

Let $G$ and $H$ be Lie groups and suppose $G$ acts smoothly on $H$ by automorphisms, i.e. there is a Lie group homomoprhism $\phi:G\times H\to H$ such that $\phi_g$ is an automorphism for all $g\in G$....
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Normaliser of $π(H)$ in $S_n$ is $π(G)$ for the regular representation π using semidirect product

I am now reading the semidirect product of groups from Dummit and Foote.I have got stuck in one question in the exercise of Dummit and Foote(Page no-186,problem no-19). Let H be a group of order $n$....
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Expressing the Lie algebra of a semidirect product of Lie groups as matrices

I am trying to work through an example of a Hamiltonian action, but I want to check my work so far on a lemma on Lie groups. Let $G$ and $H$ be Lie groups, and suppose $G$ acts on $H$ via ...
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55 views

Computing the multiplication of elements to generate the Cayley graph of a semidirect product

I have computed a semidirect product, $s$ of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ as below and have drawn a Cayley graph for $s$ with respect to a generating set $S$. But I wanted ...
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34 views

Is this an isomorphism for $D_8$?

Given the $20^{th}$ primitive root of unity $\omega$, we know that it's splitting field is $\Bbb Q(\omega)$. Furthermore we know that it's a Galois extension and that it's minimal polynomial over $\...
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22 views

Complexification of a product group

Since the complexification of $SO(4)$ $$SO(4)_\mathbb{C}\cong SL(2,\mathbb{C})\times SL(2,\mathbb{C})\tag{1}$$ it is true to say that $$\left(SU(2)\times SU(2)\right)_\mathbb{C}\cong SL(2,\mathbb{...
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18 views

K-theory of semidirect product

Given a semidirect product $G=A\rtimes B$ is there a general way to find the $K$-theory $K_0(G)$ and $K_1(G)$ of the semidirect product from $A$ and $B$?
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32 views

Affine group and semidirect product

I proved that $\mathrm{Aff}(n) \cong O(n) \rtimes \mathbb{R}^n$ and $\mathrm{Aff}(n) \cong \mathbb{R}^n \rtimes O(n)$, where $\mathrm{Aff}(n)$ is the affine group, $O(n)$ the orthogonal group, $\...
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45 views

Split Sequences. What is the Group?

See page 49 of this book. Proposition 3.22 An extension (17) splits if $N$ is complete. In fact, $G$ is then direct product of $N$ with the centralizer of $N$ in $G$. I believe (17) refers to $$...
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29 views

How to construct the homomorphism in semidirect product of $Z_3$ and $Z_{13}$?

I know that in the semidirect product of $A$ and $B$, the homomorphism $\phi:A\rightarrow Aut(B)$ should be $\phi_y(x) = yxy^{-1}$ but have no idea how to construct one for $\phi:Z_3\rightarrow Aut(Z_{...
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32 views

Semi direct product of Quaternion group with cyclic group of order p.

I am interested in knowing the semi-direct product of Quaternion group $Q_8$ with $c_p$, i.e. cyclic group of order $p$ where $p$ is a odd prime. We know that $\text{SL}_{2}(\mathbb{Z}_{3})$ is a ...