# 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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### Semidirect product of two cyclic groups

Describe all semidirect products of $C_n$ by $C_m$ (ie $C_n \rtimes C_m$) where $m,n \in \mathbb{N_+}$ Note: For the first attempt one needs to find all homomorphisms from $C_m \to U(n)$, but the ...
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### Intuition on the external Zappa–Szép product

$\newcommand{\Aut}{\operatorname{Aut}}$A classmate of mine recently posted an interesting question on Facebook. It didn't get an answer, and I couldn't get anywhere myself, so I'm hoping that someone ...
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### Finding 2 nonabelian, nonisomorphic groups of order 225

I need to find 2 nonabelian, nonisomorphic groups of order 225. Here's what I have so far: Let $G$ be a group of order 225. By Sylow's theorems, we have that $G$ contains $P_{25}$ and $P_{9}$, ...
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### Universal covering group of a semidirect product of Lie groups

I have a new relevant question. Let U, V, W be connected Lie groups, so that U is the semidirect product of V and W. By Schreier's theorem (Pontrjagin, Theorem 61), each group has a unique (up to an ...
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### Classifying Groups of Order 28

I am trying to classify groups of order 28. In the course of the problem, I am stuck in showing that three semidirect products are isomorphic to each other. In this problem, $G$ is a group of order 28,...
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### Classification of Schmidt group to $\mathscr{B}$-group.

Definition: A Schmidt group is defined to be a non-nilpotent finite group with the property that every proper subgroup is nilpotent. Also, a group is said to be a $\mathscr{B}$-group if any ...
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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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### “Super-complement” of a normal subgroup

$\DeclareMathOperator{\Aut}{Aut}$Consider a group $G$ with a normal subgroup $N\triangleleft G$ and suppose that $G$ has a subgroup $H$ so that $G = HN$. Then $H$ acts on $N$; in other words, we have ...
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### Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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### Relation between kernels of homomorphisms and the semidirect product of groups

I'm reading Abstract Algebra of Dummit and I got stuck on this point. When we try to build the semi-direct product of 2 group predefined, such as $H$ and $K$, then all the semi-direct product groups ...
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### Normal subgroup $K$ which has no complement.

Rotman's book on Group Theory claims that a normal subgroup $K\trianglelefteq G$ need not have a complement. I recall what is meant by complement. Definition. Let $K$ be a (not necessarily normal) ...
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### Groups of order 306 has 4 types of group containing an element of order 9

There are at most 4 groups of order 306 = $2 \times 3^2 \times 17$, containing an element of order 9. I want to prove the above statement. My trial is below; Let $G$ be such group. If $G$ is ...
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### Showing $\Bbb{Z}_q \rtimes Q_8$ has the presentation $\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$

Let $G \cong \Bbb{Z}_q \rtimes Q_8$. Then $G$ has a presentation as follows $$\langle x,y,z \mid x^q=y^4=z^4=[x,y]=1, y^z=y^{-1}, y^2=z^2, x^z=x^{-1} \rangle.$$ I dont understand why $x^z=x^{-1}$? ...
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### Characters of semidirect and wreath products of non-Abelian finite groups

In Section 3.2 of Linear Representations of Finite Groups by Serre, the author gives a formula to compute the characters of product of finite groups (Abelian or non-Abelian). First Serre defines the ...
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