# 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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### Classification of groups of order $18$

I was just going through the first classification- $$|G|=18=3^2\times 2$$ Then $G$ has a subgroup of order $9$(normal, say $K$) and a subgroup of order $2$ (say $H$). I want someone to help me with ...
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### How do we find the homorphism from $\mathbb{Z_2} \to{\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ [closed]

How do we find the homorphism from $\mathbb{Z_2} \to {\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})?$ I know that ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ is isomorphic to $GL_2(\mathbb{Z_3})$. We ...
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### Perfect semi direct products

Let $\pi,V$ be a representation of a perfect group $G$. I'm interested in sufficient conditions for a semi direct product like $V \rtimes_\pi G$ to be perfect. Requiring that $\pi$ is faithful ...
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### $p,q$ primes, $p\mid q-1$. Weaker assumption in the proof of the existence of non-trivial $C_p\ltimes C_q$?

Motivated by the fact that the non-existence of non-trivial $C_p\ltimes C_q$, for $p\nmid q-1$, can be proven without any piece of information on the structure of $\operatorname{Aut}(C_q)$, not even ...
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### Generalization of Semidirect Product (Finite Groups with non-trivial minimal subgroup)

Suppose any two non-trivial (non-singleton) subgroups of a group $G$ have a non-trivial intersection. $|G|$ is necessarily a prime power, because by Cauchy's theorem, for any prime $q$ dividing $|G|$, ...
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### Weyl group of $C_2$ in $C_2 \ltimes U(n)$

Let $C_2$ denote the cyclic group of order $2$ and let $C_2 \ltimes U(n)$ denote the semi-direct product of $C_2$ with the unitary group, where $C_2$ acts on $U(n)$ by complex conjugation. I want to ...
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### Is this semi-direct product residually finite?

Consider the group $G=K\rtimes \mathbb{Z}$ defined as follows: The subgroup $K$ is generated by elements $x_i,y_k$ with $i,k \in {\mathbb Z}$ and $k > 0$, and it has defining relations \begin{...
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### Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$?

Let $G= A \rtimes B$ where $A,B \leq G$. ( where $\rtimes$ denotes semi-direct product) Question- Let $N$ is normal subgroup of $A$, is $N \rtimes e$ normal subgroup of $G= A \rtimes B$ ? If yes, ...
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### Center of split extension of groups of co prime orders [closed]

Let $G = A \rtimes B,$ where $A$ is a finite $p$-group and $B$ is a finite $p'$ group. When can we say that $Z(G) \subseteq A?$ For example the center of the group $C_3 \rtimes C_4$ is $C_2$ and hence ...
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### Is there any factorization of Leibniz algebras?

A semigroup S is factorisable if there are subsemigroups A and B such that S = AB. In the case of Leibniz algebras, can we say that a Leibniz algebra is a direct sum of two subalgebras? Any reference, ...
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### Is $\operatorname{Hol}(D_4)$ isomorphic to a familiar group?

We define the holomorph of a group, $\operatorname{Hol}(G)$, as its semidirect product $G\rtimes _f\operatorname{Aut}(G)$. As it happens (as is shown here), $D_4\approx\operatorname{Aut}(D_4)$, and we ...
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### In $\operatorname{Hol}(\Bbb{Z}_{10})$, is every mapping from $G\rightarrow\operatorname{Aut}(G)$ just an identity mapping?

Consider $f: G\rightarrow\operatorname{Aut}(G)$ where $G=\Bbb{Z}_{10}$. Since this is equivalent to considering $G\rtimes_f G$, we know that $f$ is defined as $f_a(b)=aba^{-1}$. There are four ...
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### If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

I recently answered the following question: If a finitely generated semi-direct product $\mathbb{Z}$ acting on a non finitely generated group, can there be fixed points? I have a related question: Is ...
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### Characterization of the isomorphic semidirect products

Let $A$ and $G$ be two finite abelian groups and let $\alpha$, $\beta:G\rightarrow{\rm Aut}(A)$. Suppose that $\alpha (G)$ and $\beta (G)$ are conjugate subgroups of ${\rm Aut}(A)$. Are the semidirect ...
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### Obtaining splitting of quotient map from semi-direct product

Suppose $N \lhd G$. Given isomorphism $i: N \rtimes_\pi G/N \to G$ and action $\pi: G/N \times N \to N$, show how to obtain a splitting $\varphi: G/N \to G$ of the quotient map $G \to G/N$. Here, we ...
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### Information about $P'$ in $P \rtimes Q$

Let $p$ and $q$ be distinct primes. Suppose we have a group $G = P \rtimes Q,$ where $P$ is a non-abelian $p$-group and $Q$ is an abelian $q$-group such that it is a subgroup of ${\rm Aut}(P).$ Then ...
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### Generating sets of semi-direct products with $\mathbb{Z}_2$

Suppose a group $G$ splits as a semidirect product $N\rtimes\mathbb{Z}_2$, and let $\phi:G\to\mathbb{Z}_2$ the the associated quotient map. If I have a subset of elements $\{g_1,\dots,g_n,h\}$ of $G$ ...
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