# 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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### Presentation of non Abelian group (C7⋊C3)⋊C2 . [closed]

I want the Presentation of non Abelian group (C7⋊C3)⋊C2 of order 42.
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### Equivalent definition semidirect products

We were recently taught in lecture the definition of the semidirect product: Definition: A group $G$ is a semidirect product of Subgroups $H,K$ if $H$ is normal and the canonical projection $G \to G/H$...
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### $N \rtimes H$ vs $H \rtimes N$

When we define the semidirect product for $N \lhd G$ with $H < G$. We assumed $N$ is normal because that makes $f: N \times H \to G$, $f(n,h) = nh$ an isomorphism when we assume $f$ is a bijection. ...
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### Properties of Bicrossed Product multiplcation

I am reading Kassel's Quantum Groups book (the chapter on Drinfeld doubles). In it, there is the following claim: If $H,K\subseteq G$ are groups such that $\forall g\in G$, $\exists!(y,z)\in H\times K$...
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### Show that the Hyperoctahedral Group is a semi-direct product

I am confused about an exercise from Representations of Finite Groups by C.Musili (Exercise 7.10.1), which asks the reader to show that the hyperoctahedral group $B_n$ is a semi-direct product. First ...
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### What is the smallest split-simple non-simple group other than generalized quaternion groups and cyclic groups?

The quaternion group $Q_8$ is split-simple, i.e. it cannot be written as an internal semidirect product of proper subgroups. In fact all generalized quaternion groups are split-simple, as are all ...
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### Group Ring $F(G\rtimes H)$ .

Is there any relation between group rings $F(G\rtimes H)$ and $F(G)\rtimes F(H)$, where F is a finite field and $\rtimes$ is semi direct product of finite groups $G$ and $H$?
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### A problem about semi-direct product which is a direct product

I have been stuck on this problem for a while now. I tried to look up similar problems with possible hints but couldn't find anything. The problems is as follows: Let $N\rtimes_\theta H$ be the semi-...
### Do all unitary simple groups $U_{2n+1}(2)$ have maximal subgroups of the form $3^{2n}:S_{2n+1}$?
In the ATLAS, the unitary simple groups $U_5(2)$ and $U_7(2)$ have maximal subgroups of structures $3^4:S_5$ and $3^6:S_7$, respectively. It seems that they are subgroups of the generalized symmetric ...