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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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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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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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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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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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 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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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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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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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 ...
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 $\operatorname{Aut}... 1answer 57 views Subgroups of semi-direct products of two elementary abelian subgroups. [closed] First question: Let$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}...
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). ...