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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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Abelianization of $\mathbb{Z}\ltimes_\varphi \mathbb{Z}^n$

i would like to ask how to compute the abelianization of the semidirect product $\mathbb{Z}\ltimes_\varphi\mathbb{Z}^n$ where the action is $\varphi(k)v=A^k v$ where $A$ is a fixed invertible matrix ...
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$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H$ for $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, $\psi \in Aut(H)$

Let $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, if $\psi \in Aut(H)$, prove that $$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H.$$ This is mentioned in the original post. ...
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Find all group of order $20$ which is a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$

Problem: Find all group of order $20$ that are a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$. My attempt: We knew that a cyclic group of order $n$ is isomorphic ...
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Find all semidiect product of $(\mathbb{Z}_4,+)$ by $\text{C}_2 = \{{a\}}$

Problem: Find all semidiect product of $(\mathbb{Z}_4,+)$ by $\text{C}_2 = \{{a\}}$ (the cyclic group of order $2$). My attempt: We know that $(\mathbb{Z}_4,+)$ is a cyclic group of order $4$. To ...
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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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Does the interior semidirect product of Lie groups $G = N \rtimes H$ respect the projection $G \to H$?

Suppose we have some matrix Lie groups $N$ and $H$ both subsets of the $n \times n$ matrices, and a matrix Lie group $G$ for which we know $G = N H$, $N$ is a normal subgroup of $G$, and $N \cap H = \...
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Isomorphism factor by factor

Consider groups are finite. Let $G_1 = A \rtimes_{\phi_1} B_1$ and $G_2 = A_2 \rtimes_{\phi_2} B_2$. Note that $A_1,A_2,B_1,B_2$ are cyclic groups. It is also known that $A_1 \cong A_2$ and $B_1 \...
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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 $$\langle \tau\rangle \cap A_n = 1$$ $$A_n \tau = S_n$$ $$A_n \unlhd S_n$$ and thus ...
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Semidirect product of subgroups

Let $G$ be a group with $N \unlhd G$ such that there is $K \leq G$ with $KN = G$ and $K \cap N = 1$. It is well known that in this case $$G \cong N \rtimes_\phi K$$ where $\phi: K \to Aut(N): k \...
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What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
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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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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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Internal Semidirect with Factors Isomorphic to “Outside” Groups

Here is a conjecture of mine: If $G$ is the internal semidirect product of $N \unlhd G$ and $Q \le G$, and $\phi_1 : N' \to N$ and $\phi_2 : Q' \to Q$ are isomorphisms, then there is some $\theta :...
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How to show that $G$ can be expressed as a semidirect product

Let $G$ be a group of order $42$. Prove that $G$ is a semidirect product of a normal subgroup of order $21$ and $\mathbb{Z}_2$. My attempt: $G$ has unique Sylow 7 subgroup and Sylow 3 subgroup is not ...
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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 $A = B \rtimes_\beta K$, ...
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Give all groups of cardinality 12. [duplicate]

Give all groups of cardinality 12. What I did: let G be such a group. $12 = 2^2.3$ so G has a 2-Sylows $S_2$ and a 3-Sylows $S_3$. I proved that either $S_2$ or $S_3$ is normal and $S_3\cap S_2=\{...
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isomorphism between semi-direct products

given two groups H and K, and two morphisms $φ$ and $φ'$ from K to $Aut(H)$. given $σ \in Aut(K)$ such that $φ' = φ ◦ σ$, prove that $H \rtimes_φ K \cong H \rtimes_{φ'} K$. I found this simple ...
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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 $...
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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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Equivalent Conditions of Split Extension of Groups

Definition of split extension of $Q$ by $N$: An extension of $Q$ by $N$, $$1 \longrightarrow N \longrightarrow G \longrightarrow Q \longrightarrow 1,$$ is said to be split if it is isomorphic to ...
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GAP semidirect product

I am newbie in the GAP and in the group theory. Now I am trying to make semidirect product if GL(3,2) and GL(3,2) inversed and transposed. I use code below ...
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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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Is a finite centerless metabelian group always a semidirect product of two abelian groups?

Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups? It does not seem true to me, but I failed to find any counterexamples. Actually, ...
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Identification of groups given by a description

In a classification of groups of order $p^3q$, where $p, q$ are distinct primes, when considering non-nilpotent groups with normal Sylow $p$-subgroups they mention that these groups have the form $N \...
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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 $\...
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Understanding semidirect product by constructing a non-abelian group of order $21$

I just learnt semidirect product, but only know the basic definition, not gaining the true understanding of it. There is an example that asks the reader to construct a nonabelian group of order $21$. ...
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Non cyclic group of order $p^3$ satisfies $G \simeq H \rtimes_{\theta}K$

Let $G$ be a non-cyclic group of order $p^3$ for an odd prime $p$. Prove that $G \simeq H \rtimes_{\theta}K$, where $H$ is a normal subgroup of $G$ of order $p^2$, $K$ is a subgroup of order $p$, and $...
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Proving $H_1 \rtimes_{\theta_2}K \simeq H_1 \rtimes_{\theta_1}K$

Let $K=C_p$ be a cyclic group of order $p$ (prime). Let $H_1 = C_p \times C_p$, and $\theta_1,\theta_2 : K \to Aut(H_1)$ two homomorphisms. Denote $G_1 = H_1 \rtimes_{\theta_1}K$ and $G_2 = H_1 \...
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Are groups constructed using semidirect product always non-abelian? [duplicate]

When using semidirect product to construct new groups based on smaller groups, we have to define a group homomorphism from the non-normal subgroup to the group of automorphism of the normal one, i.e. ...
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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 ...
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Groups of order 56 with Sylow 2-subgroup isomorphic $Q_8$

I try to classify non-abelian groups of order $56$ with sylow $2$-subgroup isomorphic to quaterion group $Q_8$. More accurately I want to construct $2$ non-isomorphic such groups. This is an excercise ...
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Structure of semi direct product.

I want to verify that structure of group $Q_8 \rtimes C_2$, i.e. semi direct product of Quaternion group of order $8$ and $C_2 = \{1, a\}$ (cyclic group of order $2$) can be defined like: If we write ...
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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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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 ...
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Is $S_4 \times C_2$ isomorphic to $(C_2 \times C_2 \times C_2) \rtimes S_3$

Let $S_n$ denote the symmetric group on $n$ letters and $C_n$ denote the cyclic group of order $n$. Consider $(C_2 \times C_2 \times C_2) \rtimes S_3$ where $S_3$ acts on $(g_1, g_2, g_3) \in C_2 \...
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How to prove that semidirect product of $Z_{13}$ and $Z_3$ is non Abelian for a non-trivial homomorphism

The semidirect product of $Z_{13}$ and $Z_3$ is given here Finding presentation of group of order 39 as $\{x,y | x^{13} = y^3 = 1, yxy^{-1} = x^3\}$. I understand how this is arrived at but to show ...
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$(\mathbb Z/p \mathbb Z \rtimes \mathbb Z/q \mathbb Z) \times \mathbb Z/q \mathbb Z \cong\mathbb Z/p \mathbb Z \rtimes (\mathbb Z/q \mathbb Z)^2$?

Given: Let $p$ and $q$ be prime numbers such that $q$ divides $p-1$. It is well-know that there is a monomorphism $\varphi: \mathbb Z/q \mathbb Z \to Aut(\mathbb Z/p \mathbb Z)$. Define ...
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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 ...
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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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1answer
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What's an example of a finite, non-abelian, non-simple group that is *not* semidirectly reducible? [duplicate]

Say I want to classify all groups of a given order. The abelian case is completely understood by the structure theorem for finitely generated abelian groups. Assume our group is non-abelian, and we ...
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A necessary condition on semidirect products to be isomorphic

Is it true that the semidirect product groups $\mathbb{Z}\ltimes_A \mathbb{Z}^n$ and $\mathbb{Z}\ltimes_B\mathbb{Z}^n$ are isomorphic if and only if $A$ is conjugate in $\operatorname{GL}(n,\mathbb{...
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If $p <q$ (primes), how to classify the semi-direct products of $\mathbb{Z}_{q}$ by $\mathbb{Z}_{p}$?

I have solved several exercises on classifying groups and I have been wanting to generalize my results. however, I came across this problem where I know that there is no semi-direct products of $\...
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1answer
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Prove that $S_4 \cong V_4 \rtimes_\phi S_3$ for any isomorphism $\phi: S_3 \to \text{Aut}(V_4)$

Note that $\text{Aut}(V_4) \cong S_3$. I know how to prove that $S_4$ isomorphic to some semidirect product of $V_4$ and $S_3$. I know if it works for an isomphorism it works for any isomorphism. ...
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1answer
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Some preliminary concepts for Rota-Baxter algebras

I am studying Rota-Baxter Lie algebras. I do not know whether there exists the notion of free product, semi-direct product and derivation map for these types of algebras. May you introduce some papers ...
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Prove that $\exists b\in \Bbb Z$ s.t $x^{ab}=x$.(A semi Direct Product question)

Suppose $K$ is a finite cyclic group, $H$ is an arbitrary group. Consider two homomorphisms $\phi_1, \phi_2: K \to \operatorname{Aut}(H)$ s.t $\phi_1(K), \phi_2(K)$ are conjugate in $\operatorname{Aut}...
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1answer
56 views

Subgroups of semi-direct products of two elementary abelian subgroups. [closed]

First question: Let $H'$ be a subgroup of $H$ and $K'$ a subgroup of $K$. Is it true that $H'\rtimes K'$ and $H'\times K'$ are subgroup of $H\rtimes K$? Second question: Let $G=(\mathbb{Z}/p\mathbb{Z}...
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Problem related to semidirect product

I have a small question regarding the semidirect product. Consider a group $G$ which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ (internal semidirect product). ...
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Request for clarification about a computation in semidirect products

In my question, For group G which is the Semidirect product of $\mathbb{Z}_3$ and $\mathbb{Z}_7 \times \mathbb{Z}_7$, where $\mathbb{Z}_7 \times \mathbb{Z}_7$ is the normal subgroup, if z is a ...
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1answer
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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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1answer
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Action inside semidirect products

When considering a group G which is the Semidirect product of $\mathbb{Z}_3$ and $\mathbb{Z}_7 \times \mathbb{Z}_7$, where $\mathbb{Z}_7 \times \mathbb{Z}_7$ is the normal subgroup, if z is a ...