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# 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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### Faithful Action of N $\rtimes$ G on N

In short, I am trying to find a faithful action of $N \rtimes G$ on $N$, where I know that the action for the semidirect product is faithful. My first attempt was $(n, g) \cdot n'=(nn') \cdot g$, but ...
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### Complement of the kernel of a group epimorphism of a semidirect product

Currently, I am going through Chapter 25 of Character theory of finite groups by Bertram Huppert. Here the authors introduce the concept of semidirect product $G \wr H$ and the kernel of its ...
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### Understanding torsors and semidirect products of groups

I'm trying to understand the semi-direct product of groups from either a categorical or a geometric perspective and failing miserably. The four things I'm hoping will fit into a coherent picture are: ...
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### Can this matrix group be obtained from $(\mathbb R,+)$ and $(\mathbb R^*,\cdot)$?

I have stumbled upon this group of matrices in an old midterm: $$G=\{ \begin{pmatrix} a & b \\ 0 & \frac1a \\ \end{pmatrix}; a,b\in\mathbb R, a\ne0 \}.$$ The students were asked to show ...
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### Visualizing $S_3 \rtimes D_4$

I am trying to visualizing $S_3 \rtimes D_4$ following this video. Here, $S_3$ is the symmetric group over three symbols and $D_4$ is the dihedral group of order $8$. The semidirect product is defined ...
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### Inner vs outer semidirect products of $S_3$ and $D_4$

I am trying to understand the difference between the inner and outer semidirect products of the symmetric group $S_3$ and the dihedral group $D_4$ of order $8$. The products are defined here. Inner ...