# 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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### Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$

Let $n = p_1 p_2 \cdots p_k$ be the product of pairwise distinct odd primes. Let $X$ be the set of element in $\operatorname{Aut}(\mathbb Z/n\mathbb Z)$ of order $1$ or $2$. For each $\psi\in X$, the ...
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### When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
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### When does a representation of a compact Lie group extend to an extension of the group?

Let $G$ be a compact simple Lie group, let $H$ be its (finite) group of outer automorphisms. I'm interested in the semidirect product $G' = G \rtimes H$. Let $\rho : G \rightarrow U(n)$ be an $n$-...
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### Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, \mathbb{Z}/p\mathbb{Z})$

This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous ...
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$n>1$. Prove that $n$ is an abelianness-forcing number iff. $n=p_1^{a_1}p_2^{a_2}\dots p_r^{a_r}$, where $p_1,p_2,\dots,p_r$ are distinct primes, is -cubefree -$p_i\nmid p_j^{a_j}-1,\... 0answers 95 views ### Groups of order 306 has 4 types of group containing an element of order 9 There are at most 4 groups of order 306 =$2 \times 3^2 \times 17$, containing an element of order 9. I want to prove the above statement. My trial is below; Let$G$be such group. If$G$is ... 0answers 50 views ### Showing$\Bbb{Z}_q \rtimes Q_8$has the presentation$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$Let$G \cong \Bbb{Z}_q \rtimes Q_8$. Then$G$has a presentation as follows $$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$$ I dont understand why$x^z=x^{-1}$? ... 0answers 228 views ### Characters of semidirect and wreath products of non-Abelian finite groups In Section 3.2 of Linear Representations of Finite Groups by Serre, the author gives a formula to compute the characters of product of finite groups (Abelian or non-Abelian). First Serre defines the ... 0answers 412 views ### On monomial matrices (Generalized Permutation Matrices ) A matrix$a\in GL_{n}(F)$is said to be monomial if each row and column has exactly one non-zero entry. Let$N$denote the set of all monomial matrices. I have already proved here that the ... 0answers 101 views ### How to write$\Bbb Z_q\rtimes\Bbb Z_p$as$\langle a,b\mid a^p=b^q=1,aba^{-1}=b^{i_0}\rangle?$I am referring specifically to this example http://planetmath.org/groupsoforderpq In Case 2, the group should be$\mathbb{Z}_q\rtimes\mathbb{Z}_p$. How do we write it in the presentation of $$G=\... 0answers 31 views ### Is it true that if bk\in Z(B\rtimes K) then b\in Z(B)? Is this fact true? Let be G= B \rtimes K and I suppose that w=bk \in Z(G), with b\in B and k \in K. Is it then true that b\in Z(B)? I have tried to prove it but I didn't succeed. Thanks! 0answers 143 views ### Wreath product of subgroup with symmetric group I am going through Generic Quantum Fourier Transforms by Moore et al. I would like to put the screenshot of the section I am confused about below. So, Why does H need to be of size poly(n) for ... 0answers 385 views ### Direct product, semidirect product and associativity I know that the direct product operation is associative, and that in general the semidirect product operation is not associative. But what about when we work with a mix of both? I'm trying to ... 1answer 115 views ### p-Sylow conjugation in semidirect product I have a semidirect product group G=N\rtimes H, where N is a normal p-Sylow subgroup of G of order p^d and H has order m. Additionally, m and p are coprime and the centralizer of N ... 0answers 59 views ### Metaplectic group as semidirect product There is a homomorphism$$\varphi : \{\pm 1\} \longrightarrow \mathrm{Aut}(SL_2(\mathbb{R}))$$defined by \varphi(-1)(A) = (A^{-1})^T and this allows us to construct the semidirect product \{\pm 1\}... 0answers 187 views ### Conditions for a group to be isomorphic to semidirect product of its subgroups Let G a group and N a normal subgroup of G. If G it have a subgroup H s.t. H \cap N is the trivial subgroup and H is isomorphic to G/ N then G is isomorphic to N\rtimes H. Could ... 0answers 236 views ### Adjoint action of semi-direct product Let G and H be Lie groups with associated Lie algebras \mathfrak{g}:=\text{Lie}(G) and \mathfrak{h}:=\text{Lie}(H) and adjoint actions \text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g}) ... 0answers 49 views ### Is there a sort of “two-sided semidirect product”? [duplicate] Let G,H be groups. Suppose we have both an action of G on H, and an action of H on G, both non-trivial. Let "\cdot" define the former action, and \circ define the latter. What can we ... 1answer 82 views ### What are the semi-direct products of \mathbb{Z} with itself? (Check my work please) I am just starting out with semi-direct products. I would like to list and describe the semi-direct products of \mathbb{Z} with itself. I first need to find the automorphisms \varphi from \... 1answer 33 views ### Is \langle A,G\rangle=G\rtimes_{\sigma}A? Given groups A and G; suppose A act via automorphism on G, i.e. there exists \sigma:A\rightarrow\operatorname{Aut}G, a\mapsto\sigma_a:g\mapsto g^a. From this action we can define a ... 2answers 141 views ### If p and q are distinct primes, construct all semidirect products of \mathbb{Z}_p by \mathbb{Z}_q. This is exercise 7.29 in Rotman's book "An Introduction to the Theory of Groups". If p and q are distinct primes, construct all semidirect products of \mathbb{Z}_p by \mathbb{Z}_q. I don't ... 2answers 101 views ### Can this matrix group be obtained from (\mathbb R,+) and (\mathbb R^*,\cdot)? I have stumbled upon this group of matrices in an old midterm:$$G=\{ \begin{pmatrix} a & b \\ 0 & \frac1a \\ \end{pmatrix}; a,b\in\mathbb R, a\ne0 \}.$$The students were asked to show ... 3answers 133 views ### Find a non-trivial semidirect decomposition of the following groups Find a non-trivial semidirect decomposition of the groups S_n, n \geq 3, D_{2n}, n \geq 3 and A_4. Prove that A_n, n \geq 5 and Q_8 have no non-trivial semidirect decompositions. How ... 1answer 26 views ### Find all semidirect products of (\mathbb{Z}_4,+) by C_2 Problem: Find all semidirect products of (\mathbb{Z}_4,+) by C_2 (the cyclic group of order 2). My attempt: We know that (\mathbb{Z}_4,+) is a cyclic group of order 4. To find all ... 1answer 41 views ### Question regarding possiblity for existence of a particular semidirect product Can there be semidirect products (\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q having p <q? I've seen this group for p>q values but not for p<q values, therefore can ... 1answer 76 views ### Understanding semidirect product for group of order 30 I am using Dummit and Foote (pg 181-182) and trying to understand section 5.5 on semidirect product for a group of order 30. They start out with a reminder that if H is of order 15 it is a normal ... 1answer 84 views ### Semidirect product command in GAP [duplicate] I need to type the group in GAP. Is it correct if I type as, "SemidirectProduct(ZmodnZ(3),AbelianGroup([7,7]));"? Will it recognize that AbelianGroup([7,7]) is the normal subgroup? 3answers 81 views ### Fundamental group of a quotient of a cylinder Let X be a connected topological space which admits a universal covering. Let Y be the topological space obtained as following:we have an homeomorphism f : X \to X , we take X \times \left[0,... 1answer 167 views ### Interpretation of wreath products in general and on symmetric groups I've been trying to study how wreath products work. In several textbooks, K wr H is defined as the semidirect product of H acting on the set of all functions from X to K. Now, in my understanding, the ... 1answer 48 views ### Online reference about semi-direct products in finite group theory? The title says it all: Do you know a comprehensive (preferably online) reference about semi-direct products in (finite) group theory? I would like to know much more about semi-direct products in the ... 1answer 134 views ### A necessary condition for two semi-direct products to be isomorphic. Notations and definitions. For all A\in\textrm{GL}_n(\mathbb{Z}), A is hyperbolic if and only if none of its complex eigenvalue has module 1. For all (A,B)\in\textrm{GL}_n(\mathbb{Z})^2, one ... 1answer 103 views ### Proof that G is the semi direct product of P and Q if and only if the composition \phi\circ \iota : P\rightarrow G/Q is an isomorphism Let G be a group with subgroups P and Q. Assume Q is normal. Define G to be the semi direct product of P and Q if PQ=\{pq: p\in P,q\in Q\}=G and P\cap Q=\{e\}, where e is the ... 2answers 157 views ### When does a p-group split as semidirect product with its commutator? Given a (finite) group G and its commutator subgroup G', when is it the case that$$1\to G'\to G\to G/G'\to 1$$splits? Specifically, can we say anything if we add the assumption that G' is ... 2answers 932 views ### Prove that the Dihedral group D_n is isomorphic to Z_n \rtimes_{\psi} Z_2 I consider the following map \psi : Z_2 \rightarrow Aut(Z_n) where we map the identity element 0 to the identity map and 1 \mapsto \theta : Z_n \rightarrow Z_n where \theta(x) = -x. I am not ... 1answer 57 views ### Extension of \mathbb Z_2 by SO(n) How to show that the extension of group \mathbb Z_{2} by \operatorname{SO}(n):$$\operatorname{Id} \to \operatorname{SO}(n) \to \operatorname{O}(n) \xrightarrow{\det} \mathbb Z_2 \to 1$$is a ... 1answer 234 views ### The normaliser of the left regular image [D&F] I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 186): Let H be a group of order n, let K=\text{Aut}(H) and form G=\text{Hol}(H)=H \rtimes K (where ... 2answers 176 views ### Help with semidirect product I need help with this problem, i am trying to understand the semidirect product, so if anyine could help or give me some ideas Let G be the group generated by <a,b> and the relations aba^{-... 1answer 41 views ### Show that S_n \cong A_n \rtimes C_2 [duplicate] I want to show that S_n \cong A_n \rtimes C_2. Take a transposition \tau \notin A_n. Then it is clear that$$\langle \tau\rangle \cap A_n = 1A_n \tau = S_nA_n \unlhd S_n$$and thus ... 1answer 28 views ### Behaviour of restrictions of automorphisms of groups on characteristic subgroup under epimorphisms Let$G = H \rtimes_\alpha K$, where$H$is abelian and characteristic in$G$. Let$\phi\in\mathrm{Aut}(G)$, and$\phi'$is its restriction:$\phi'=\phi\big\rvert_H$. Let$A = B \rtimes_\beta K$, ... 1answer 55 views ### A question regarding groups of order$p^2qr$When considering finite groups$G$of order,$|G|=p^2qr$, where$p,q,r$are distinct primes, let$F$be a Fitting subgroup of$G$. Then$F$and$G/F$are both non-trivial and$G/F$acts faithfully on$...
As we all know, the weyl group of lie algebra of $B_{2}$ type is $\left\{s_{1},s_{2}|s_{1}^{2}=1, s_{2}^{2}=1, (s_{1}s_{2})^{4}=1\right\}$. How can we identify this with $Z^{2}_{2}\rtimes S_{2}$? If ...