# 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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### Show that $S_n \cong A_n \rtimes C_2$ [duplicate]

I want to show that $S_n \cong A_n \rtimes C_2$. Take a transposition $\tau \notin A_n$. Then it is clear that $$\langle \tau\rangle \cap A_n = 1$$ $$A_n \tau = S_n$$ $$A_n \unlhd S_n$$ and thus ...
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### How to show that $G$ can be expressed as a semidirect product

Let $G$ be a group of order $42$. Prove that $G$ is a semidirect product of a normal subgroup of order $21$ and $\mathbb{Z}_2$. My attempt: $G$ has unique Sylow 7 subgroup and Sylow 3 subgroup is not ...
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### Split extension of groups and semidirect product

I am studing Ext functor and have some basic problem. For every semidirect product $G$ of groups $N$ and $H$, short exact sequence $0 \to N \to G \to H \to 0$ splits. On the other hand, every ...
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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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### semidirect product of isometry group

I am doing exercise about semidirect product. Here is the question: Prove that the isometry group of Euclidean space $R^n$ is $O(n)\rtimes R^n$. I was stucking. Any ideas?
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### Understanding semidirect product by constructing a non-abelian group of order $21$

I just learnt semidirect product, but only know the basic definition, not gaining the true understanding of it. There is an example that asks the reader to construct a nonabelian group of order $21$. ...
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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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### In what case the semi direct product is abelian?

Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian?
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### Are groups constructed using semidirect product always non-abelian? [duplicate]

When using semidirect product to construct new groups based on smaller groups, we have to define a group homomorphism from the non-normal subgroup to the group of automorphism of the normal one, i.e. ...
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### Question regarding possiblity for existence of a particular semidirect product

Can there be semidirect products $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$ having $p <q$? I've seen this group for $p>q$ values but not for $p<q$ values, therefore can ...
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### Groups of order 56 with Sylow 2-subgroup isomorphic $Q_8$

I try to classify non-abelian groups of order $56$ with sylow $2$-subgroup isomorphic to quaterion group $Q_8$. More accurately I want to construct $2$ non-isomorphic such groups. This is an excercise ...
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### Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
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### Structure of semi direct product.

I want to verify that structure of group $Q_8 \rtimes C_2$, i.e. semi direct product of Quaternion group of order $8$ and $C_2 = \{1, a\}$ (cyclic group of order $2$) can be defined like: If we write ...
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