# 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 products of 3 groups?

When we have a group $A$ acting on a group $B$ it is possible to construct a third group "containing" both $A$ and $B$ where the action of $A$ on $B$ coincids with the conjugation of the ...
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### For $N\unlhd G$ for $\mathfrak{B}$-group $G$ with $G/N$ a free $\mathfrak{B}$-group, show $\exists H\le G$ with $G=HN$ and $H\cap N=1$

This is Exercise 2.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". In the book, maps are evaluated from left to right. According to Approach0, the exercise is new to ...
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### Is Symmetric group on 5 symbols is the semi-direct product?

Is the Symmetric group on 5 symbols is the semi-direct product of groups $A_5$ and $C_2$, i.e. $$S_5\cong A_5\rtimes C_2?$$ Here $A_5$ is considered as a normal subgroup. Please help.
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### Role of semidirect product and intuition for $O(2)\simeq U(1)\rtimes \mathbb{Z}_2$?

I have a very basic understanding of some common groups, and I'm trying to get some intuition for this isomorphism. My thinking so far is that $O(2)$ is rotations and reflections in $\mathbb{R}_2$, ...
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### $O(n)\cong SO(n)\rtimes O(1)$

I want to prove that $O(n)\cong SO(n)\rtimes O(1)$ as Lie groups. I have the following result: If $G,N,H$ are Lie groups, then $G\cong N\rtimes H$ iff there are Lie group homomorphisms $\phi:G\to H$ ...
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### Is the (outer) semidirect product, $N \rtimes H$, ever characterized as a group of bijections on a group containing $N$?

I'm an undergraduate who was introduced to semi-direct products of groups, $N \rtimes H$, relatively recently. At first, I found the multiplication rule for (outer) semi-direct products very annoying. ...
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### Identify subgroup of a semidirect product in GAP

I have created an external semidirect product of two groups $H$ and $K$ in GAP. Now, I want to identify $H$ as a subgroup of this semidirect product, but I am unable to identify $H$. How can I ...
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