# 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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### How many $3$-Sylow subgroups are there in $\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$ with $|\ker\rho| = 2$?

Let $G=\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$ with $|\ker\rho| = 2$. How many $3$-Sylow subgroups are there in $G$? I know that the number is $1$ or $7$, but I'm stuck.
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### Automorphism group of elliptic curve in char 2

I'm trying to calculate the automorphism group of elliptic curve with $j$-invariant $0$ in a field $K$ of characteristic $2$. Let $Y^2Z+b_3YZ^2=X^3$ the elliptic curve. The substitutions preserving ...
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### $S_n \cong A_n \rtimes \{1,-1\}$

I have seen this question already a few times, but I still do not understand the actual answer. I have to prove the isomorphism between $S_n$ and the semidirect product of $\{1,-1\}$ and $A_n$. I am a ...
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### Structure of non abelian finite p-groups

I am familiar with the concepts of direct products, semi-direct products, wreath products and central products of groups. After seeing the classification of finite $p$-groups upto order $p^4$,(Theory ...
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### Quotient of $\mathbb R^n\rtimes O\left(n\right)$ by $\mathbb Z^n\rtimes D_8$

What is the coset space $\frac{\mathbb R^n\rtimes O\left(n\right)}{\mathbb Z^n\rtimes D_8}$ as a manifold? I saw a claim that it is the $n$-dimensional torus $\mathbb T^n=S^1\times\cdots\times S^1$? ...
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### Split extension of finite group and Sylow subgroup by $p$-group

Let $p$ be a prime and let $A$ be an abelian normal $p$-subgroup of a finite group $G$. Hence, for any Sylow $p$-subgroup $H$ of $G$, it holds that $A$ is contained in $H$ and so $A$ is a normal ...
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### Dihedral group as a semidirect product?

It is known that the dihedral group $D_{2n}$ is isomorphic to the semidirect product $Z_n\rtimes Z_2$, where both $Z_n,Z_2$ are cyclic. My question is, for a semidirect prouct, the two subgroups ...
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### Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2$ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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### If $G$ acts on $V$, how does $G^n\rtimes S_n$ act on $V^{\otimes n}$?

In a paper it was taken as obvious that if a finite group $G$ acts on a vector space $V$, then the semidirect product $G^n\rtimes S_n$ acts on $V^{\otimes n}$. I've tried to elaborate on how I think ...
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### Representing a group as a quotient of a free group

Consider $G=F \rtimes T$, where $F=\mathbb{Z}_3 \times \mathbb{Z}_3$ and $T=\mathbb{Z}_5$. Let $\phi : \mathbb{Z}_5 \rightarrow Aut(\mathbb{Z}_3 \times \mathbb{Z}_3)$. It is said that any group is ...
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### $M<K\rtimes H$ is a semidirect product?

Let $H,K$ be two finite groups, $K$ abelian, and let $M$ be a subgroup of $K\rtimes H$. Consider the projection $\pi:K\rtimes H \rightarrow H$ on the Second factor. Let us suppose that $\pi(M)=H$. ...
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### Closed-form exponential and logarithmic map of Galilei group

I have a question regarding the logarithmic map $log: G\mapsto \mathfrak{g}$ and exponential map $exp: \mathfrak{g}\mapsto G$ between the Galilei group and its Lie algebra. The Galilei group of two ...
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### Determining the homomorphism $\varphi: H \to \mathrm{Aut}(K)$ given a section $s: H \to G$

Per Wikipedia, a split extension is an extension $$1 \to K \overset{\beta}{\to} G \overset{\alpha}{\to} H \to 1$$ with a homomorphism $s: H \to G$ such that going from $H$ to $G$ by $s$ and then back ...
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### Why does every $\varphi: K \to \mathrm{Out}(H)$ determine an unique extension of $H$ by $K$ when $Z(H) = 1$?

Every homomorphism $\varphi: K \to \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. Why is this true for groups $H$ with a trivial center? Even if we only consider split extensions, as ...
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### 2 cocycle and semidirect products

I am trying to solve the following question: Let (K, +) be an abelian group and let H be a group that acts on K by (group) automorphisms (k1, h1) • (k2, h2) = (k1 + h1 · k2 + ε(h1, h2), h1h2) Gε ...
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### Generators of $C_3\rtimes C_2$

Can I write elements of $G=C_3\rtimes C_2$ as $$\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1)\}?$$ Then, what are the generators of $G$? $(0,1)$ and $(1,0)$? I've learned that the multiplication of semi-...
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### Elements of $C_3\rtimes C_2$ not $S_3$ or $D_3$

Can we represent an element of $G=C_3\rtimes C_2$ as $(a,b)$ like we do in the direct product? Because when I draw a Cayley diagram of $G$, I don't know how to label each node and arrow without the ...
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### On the number of subgroups of semidirect product

Let $G=H\rtimes K$ be the semidirect product of the normal subgroup $H$ and the subgroup $K$. Let $D$ be a subgroup of $G$ such that $K\le D$. Does $D= (D\cap H)\rtimes K$?. I believe that is true, ...
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### The natural action of $SL_2(\Bbb R)$ on $\Bbb R^2$

Suppose we have special linear group $SL_2(R)$, we can construct a semi-direct product $\Bbb R^2 \rtimes SL_2(\Bbb R)$ through the natural action of $SL_2(\Bbb R)$ on $\Bbb R^2$. What is the ...
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### Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product.

Question: Prove that $G=$SL$(2, \mathbb{F}_5)$ is an extension of $\mathbb{Z}_2$ by $A_5$ which is not a semidirect product. (This is a question from Rotman's Advanced Modern Algebra which I am ...
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### Groups which cannot be written as semidirect products

I am reading Dummit & Foote, Abstract Algebra, 3e, p.103ff. We know that the first part of Jordan-Hölder program, the classification of finite simple groups, are finished. But it is not written ...
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### Number of subgroups of semidirect product of two elementary abelian subgroups

It is well known that a subgroup of the semidirect product $H\rtimes K$ is not in general semidirect product of two subgroups $H'\le H$ and $K'\le K$ but always exist some subgroups of $H\rtimes K$ on ...
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### Group action by a semi-direct product.

Let $G$ be a group and $H,N$ subgroups, of which $N$ is a normal subgroup. Suppose that $G= H \ltimes N$ and that $H \cap N = 0$. Is any action of $G$ on a set $X$ equivalent to an action of $N$ on ...
### Presentation $\langle x,y \mid x^3=y^3=(xy)^3=1\rangle\cong\langle t\rangle\ltimes A$
Hi: This question has already been answered here: Show $\langle x,y|x^3=y^3=(xy)^3=1\rangle$ is isomophic to $A\rtimes\langle t\rangle$, where $t^3=1$ and $A=\langle a\rangle\times\langle b\rangle$. ...