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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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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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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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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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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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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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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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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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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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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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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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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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Semi-direct v.s. Direct products

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct ...
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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 extension of groups and semidirect product

I am studing Ext functor and have some basic problem. For every semidirect product $G$ of groups $N$ and $H$, short exact sequence $0 \to N \to G \to H \to 0$ splits. On the other hand, every ...
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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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Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$

When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following: $\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$ See https://groupprops....
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Holomorph of a group $G$, then the automorphism of $G$ are inner automorphisms

In my course notes of algebra it says: Let $G$ be a group. Then $\mathrm{Aut}(G)$ acts on $G$ in a natural way through automorphisms. This allows us to consider $A:= G \rtimes \mathrm{Aut}(G)$. In ...
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The holomorph of $Z_2 \times Z_2$

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184, Exercise 5): Let $G=\text{Hol}(Z_2 \times Z_2)$ (a) Prove that $G=H \rtimes K$ where $H=Z_2 \...
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Automorphism group of Hol($\mathbb{Z_n}$)

I am reading about semidirect product in Dummit,Foots. I wonder the automorphism group of $Hol(\mathbb{Z}_n)$, that is, $\mathbb{Z}_n\rtimes\mathbb{Z}_n^×$ with an operation $(a,b)*(c,d)=(a+bc,bd)$. ...
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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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$G\rtimes_\phi H \cong G\rtimes_\psi H$ when certain automorphisms exist

I'm reviewing for exams and came across this problem from an older exam: Let $G$ and $H$ be groups and let $\phi,\psi : H\to \operatorname{Aut}(G)$ be homomorphisms from $H$ into the group of ...
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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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1answer
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How many non-abelian groups of order $lpq$ are there?

If $l,p,q$ are primes with $l<p<q$, such that $$p\nmid (q-1)\hspace{1cm} l\mid (p-1)\hspace{1cm} l\mid (q-1) $$ I want to show that there are at least $1$ and at most $(l+1)$ non-...
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Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
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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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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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Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$

Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Lets say the generators of ...
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In what case the semi direct product is abelian?

Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian?
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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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1answer
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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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1answer
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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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1answer
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Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
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1answer
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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 ...