# 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 “bluffer's group” called?

Today I attended a course on Geometric Group Theory in Spanish, and we saw an example of a group which could be literally translated as "bluffer's group", because there is a funny way to interpret it. ...
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### An upper bound for the number of semidirect products of a finite group up to isomorphism

Given a group $G$ with identity element 1, a subgroup $H$, and a normal subgroup $N$ of $G$; $G$ is called the semidirect product of $N$ and $H$, written $G = N\rtimes H$ , if $G = NH$ and $H\cap N=1$...
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### Product of two elements in a semidirect product with distinct prime powers

Recall the definition of semidirect product: Let $G, H$ be groups and $\phi:H\longrightarrow \text{Aut}(G)$ a group homomorphism. We define the semidirect product of $G$ and $H$ ($G\rtimes_\phi H$) ...
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### Why $\int_{K}f(x,k)\pi_{m}(k^{-1}) dk=g(x)$?

Let $\tilde{\mathbb G}=G \rtimes K$ the semi-direct product of a localement compact group $G$ and a compact group $K$. Let $\pi_{m}$ is a character of the irreducible unitary representations of $K$. ...
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### Can anyone help me prove that the following statement implies G finite group is directly indecomposable? [closed]

Statement: If for any K1 and K2 groups, K1×K2 has an isomorphic subgroup to G, then K1 has a subgroup isomorphic to G and K2 also has a subgroup isomorphic to G. I know that an example when this ...
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### Determine all homomorphisms $\phi: K\rightarrow Aut(H)$.

Let $H$ be cyclic group of order 7 and let $K$ be cyclic of order 3. (a) Determine all homomorphisms $\phi: K\rightarrow Aut(H)$. My attempt: Since $K$ is cyclic, any one of the two generator of $K$ ...
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### Show that $D_3\times_\rho\mathbb{Z}_2$ is not isomorphic to $A_4$

This is the third part of a problem. I will list here the first two parts as a reference and then my attempt to solve the third one. I want to verify if the first and second are right and some help ...
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### Representations of the generalized quaternions

Given the generalized quaternion group $Q_{2^t} = \langle a,b : a^{2^{t-1} } = e, b^2 = a^{2^{t-2}}, b^{-1}ab = a^{-1}\rangle$, I have been asked to find it's (linear complex) representations. My ...
### Why is the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ $16$?
Currently, I am reading Representation Theory of Semi-Direct Products by Reyes. In section $6$, the author mentions that the number of the conjugacy classes of $D_4 \rtimes Aut(D_4)$ is $16$. My ...