# 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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### Splitting lemma for Non-abelian groups (semi-direct product)

I was reading the wikipedia page to learn about how the splitting lemma partially holds for non-abelian groups. Splitting lemma partially true $$0\to A \to B \to C \to 0$$ If a short exact sequence ...
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### Is every group the semidirect product of its center and inner automorphism group?

For every group $G$, we have $$G/Z(G)\simeq \operatorname{Inn}(G).$$ I wonder whether the quotient projection has a right inverse. I suspect it doesn’t have one in general. But I can’t find a counter ...
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### What is the semidirect product we use in Levi Decomposition

So using the Levi Decomposition for any lie algebra $\mathfrak{g}$, there exists a semisimple subalgebra $\mathfrak{s}$ such that: $Rad(\mathfrak{g})$$\ltimes$$\mathfrak{s}$=$\mathfrak{g}$ However in ...
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### How fast does the number of "fixed" points grow compared the the size of the ball in the following group?

Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $V^{(i)} \cong \mathbb Z ^5$. Let $e_1, e_2, e_3, e_4, e_5$ be the standard basis of $\mathbb Z^5$. Let $e_j^{(i)}$ be the element in $M$ ...
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### Non-Simple, Centerless Group With Exactly One Non-Trivial Normal Subgroup

Question. (Is this statement true?) Given non-simple, centerless group $G$ such that there exists exactly one non-trivial normal subgroup $N\triangleleft G$, then $G/N$ must be isomorphic to some ...
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### Trouble with understanding classifying groups with semi direct products

I'm trying to understand the following strategy on classifying groups of a particular order from Dummit & Foote's Abstract Algebra (p.181): Let $G$ be a group of order $n$. You find proper ...
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### Relation between centerless groups and semidirect products

Let $H$ be a group, $G \triangleleft H$ a normal subgroup and $K \leqslant H$ a subgroup. We say $H$ to be a semidirect product of $G$ and $K$, denoted by $H= G \rtimes K$, if $H=GK$ and $G \cap K=1$. ...
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### Alternative definitions outer semidirect product [closed]

I am working with outer semidirect products and have come across two definitions that are slightly different. Let $N,H$ be two groups and $\theta: H \to \text{Aut}(N)$ a group morphism. Then in both ...
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### Classify groups of order 20. [duplicate]

A group of order $20$ must be either a semidirect product of $\Bbb Z/4\Bbb Z$ with $\Bbb Z/5\Bbb Z$ or a semidirect product of $\Bbb Z/2\Bbb Z\times \Bbb Z/2\Bbb Z$ with $\Bbb Z/5\Bbb Z$. There are $4$...
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### Conjugate images induce isomorphic semi-direct products?

See here for the question context. Let $H_5 = \Bbb Z_5\times \Bbb Z_5$. I am trying to prove that for all non-trivial homomorphism $$\varphi:\Bbb Z_3\to \operatorname{Aut}(H_5),$$ the resulting semi-...
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Let $E$ be a linear space and $V$ a linear subspace. No assumption is made on the dimension of $E$. We write $G = \{g \in GL(E) \mid g(V) = V\}$ where $GL(E)$ is the group of invertible linear maps of ...