# 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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### Semi-direct v.s. Direct products

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct ...
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### What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
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### GAP semidirect product algorithm

Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group? EXAMPLE: ...
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### Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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### Is a semidirect product of linear groups a linear group?

It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product? Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, ...
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### $(\mathbb Z/p \mathbb Z \rtimes \mathbb Z/q \mathbb Z) \times \mathbb Z/q \mathbb Z \cong\mathbb Z/p \mathbb Z \rtimes (\mathbb Z/q \mathbb Z)^2$?

Given: Let $p$ and $q$ be prime numbers such that $q$ divides $p-1$. It is well-know that there is a monomorphism $\varphi: \mathbb Z/q \mathbb Z \to Aut(\mathbb Z/p \mathbb Z)$. Define ...
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### Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$?

Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$? I can see that $\varphi(K) = \varphi(\phi(K))$, but it is not clear to me how the ...
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### Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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### Classify groups of order $pq^2$ using semidirect product

I am struggling with semidirect products and how they can be used to classify groups of a certain order. In particular, I need help with the nonabelian case. This is the problem I am working with.. ...
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### Projection of a normal subgroup in semidirect product

Consider a semidirect product $N\rtimes G$. Consider the projection map $\pi_N\colon N\rtimes G\to N$. Suppose $\Gamma\unlhd N\rtimes G$ is a normal subgroup and that $\pi_N[\Gamma]=H$ is a subgroup ...
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### Why is the symmetry group $S_3$ not the direct product of two nontrivial groups?
I know that $S_3$ is the semidirect product of $\bigl\langle(1\ \ 2\ \ 3)\bigr\rangle \rtimes\bigl\langle(1\ \ 2)\bigr\rangle$, and I'm not sure where exactly the direct product property fails. Is it ...