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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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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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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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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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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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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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 $G\cong K\times_{\... 0answers 19 views Computing the characters of$\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$... 0answers 151 views Why$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 ... 0answers 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 \... 1answer 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? 0answers 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 ... 0answers 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 ... 0answers 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 \... 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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\... 0answers 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, $$\mathbb{Z}^n_2 \wr \mathbb{Z}_2 =... 0answers 49 views Progressed A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery: Let H be a group and K = \langle x\rangle be a cyclic group (... 0answers 141 views Borel subgroup is a semidirect product of the subgroup of all unitriangular matrices and the maximal split torus Suppose G = GL(n,q) and B\leqslant G is the Borel subgroup, N \leqslant G is the subgroup of all monomial matrices and T=B\cap N is a maximal split torus. I am trying to understand why B=U \... 1answer 69 views Show a group is a semi direct product Consider a set G of n \times n matrices with entries \{0,1,-1\} that have exactly one non zero entry in each row a column. Show G is a group and that G is the semi direct product of the ... 0answers 62 views Prove the following two groups are isomorphic I'm looking for an answer to the following problem, any responses would be greatly appreciated as I think it's quite complicated!. Let U=\mathbb{F}_{q}^{m} and W=\mathbb{F}_{q}^{n} be two vector ... 0answers 44 views Semidirect product. I have a problem with representation of this : (D_n \times D_n) \rtimes \mathbb{Z}_2, where \mathbb{Z}_2 acts on D_n \times D_n by exchanging the two components. D_n = \langle x, \ y \ | \ x^n ... 0answers 65 views Show that the group of permutations of \{1,2,3,4\} is equal to the product of it's subgroups… Show that the group of permutations of \{1,2,3,4\}$$\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 ... 0answers 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 ... 0answers 74 views 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. 0answers 123 views 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 ... 0answers 46 views 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 ... 0answers 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 ... 0answers 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 ... 0answers 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.... 0answers 7 views 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.... 0answers 27 views 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 ... 0answers 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 ... 0answers 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 \... 0answers 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{... 0answers 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$B$? 0answers 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,$\...
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 ...