Questions tagged [group-extensions]

This group is for questions relating to "group extensions", a general means of describing a group in terms of a particular normal subgroup and quotient group.

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Understanding the construction of group extension

Definition: Suppose $G$ is a group with normal subgroup $H$ and that $G/H\cong K$ then $G$ is an extension of $H$ by $K$ Let $\phi$ be an isomorphism of $G/H$ onto $K$. Let $X$ be a left transversal ...
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Non-abelian extensions of the Klein four-group $K_4$ by $\mathbb{Z}_2$

We know that we have eight extensions of $K_4$ by $\mathbb{Z}_2$ classified by $H^2_{grp}(K_4,\mathbb{Z}_2)$, see for example this link. Four of these extensions are abelian, and four of them are non-...
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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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How many $\mathbb{Z}_2$ extensions of alternating groups $A_n$ are there?

By a $\mathbb{Z}_2$ extension of $A_n$, I mean the following short exact sequence: $1 \longrightarrow A_n \longrightarrow G \longrightarrow \mathbb{Z}_2 \longrightarrow 1$. Question: How many $G$ are ...
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What is the free product with amalgamation with the trivial group?

I am working on algebraic topology. I am trying to prove the Wirtinger presentation using the Van Kampen theorem. However, I have some difficulties understanding the concept of free product with ...
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HNN-extension and Centralizer

I am currently studying the book of Graham Higman and Elizabeth Scott, The Existentially Closed Groups, London Mathematical Society Monographs New Series, Clarendon Press Oxford, 1988. In the Section ...
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How to construct the action on chains of the Bockstein homomorphism for homology

Given a manifold $X$ and short exact sequence of abelian groups $$1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1$$ we get the Bockstein map in cohomology ...
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Natural group extension constructed from Schur cover and its outer automorphism group

Let $S$ be a finite (non-abelian) simple group. Then there always exists a natural extension of $S$ by the outer automorphism group $Out(S)$ with elements of $Out(S)$ acting as outer ...
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Nontrivial extension of cyclic group by simple group

Let $G$ be a (non-abelian) finite simple group. An extension $G\cdot m$ is nontrivial if it is not isomorphic to the direct product $G \times m$. Suppose that there exists a nontrivial extension ...
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Consider a finite abelian group $G$ and a subgroup $H\subset G$, and denote by $A=G/H$ the quotient. Then $G$ is an extension of $A$ by $H$ determined by the short exact sequence $$1\rightarrow H\... 0 votes 0 answers 31 views Internal symmetries in abelian or non abelian groups so I've been studying the centrally extended Galilei group, by restricting to 1D translation, the group becomes abelian and that means that all the irreps of the groups are of dimension 1. A ... • 59 -1 votes 1 answer 70 views Split extension definition I have the following definition of a split extension: An extension G of H by N is called a split extension if the canonical projection \pi:G \to G/N (G/N is isomorphic with H) has a ... • 191 8 votes 1 answer 180 views Does every extension of a finite group by \mathbb{R}^n split? Suppose G is a topological group containing a closed normal subgroup N isomorphic to (\mathbb{R}^n, +) such that G/N is finite. Is N a semidirect factor? Equivalently, does G contain a ... • 133 3 votes 1 answer 95 views Central extension of perfect group has perfect commutator subgroup Suppose that  G/Z(G)  is perfect. Can we conclude that the commutator subgroup  [G,G]  is perfect? I think that  [G,G]  must be a perfect central extension of  G/Z(G)  and that  G  has the ... • 7,950 1 vote 1 answer 111 views When is  N(T)  a semidirect product? Direct product?  \DeclareMathOperator\SO{SO} \DeclareMathOperator\SU{SU} For which compact connected simple lie groups  G  does the sequence$$ 1 \to T \to N(T) \to W \to 1 $$split? Here  T  is the maximal ... • 7,950 1 vote 0 answers 81 views A doubt in a theorem about group extensions I am reading this chapter. Part 1 in theorem 1.2 tells us that Let E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1 be an extension of A by G let t : G \to X be a section of f i.e. t(... • 5,339 3 votes 0 answers 137 views Is a profinite HNN-extension of the profinite completion of a group the profinite completion of an HNN-extension of the same group? I have a (maybe silly) question: A profinite HNN-extension H(G) of a profinite group G is defined as the profinite completion of an abstract HNN-extension of G. Here is my question. If G = \... • 4,015 3 votes 1 answer 202 views In GAP, how to to construct all non-split extensions? It is well explained how to construct split extensions (semidirect products) in the GAP manual; however, for the non-split extensions, I couldn't find any method or a source of code implemented to ... • 535 3 votes 0 answers 35 views Does the class of groups with all non-abelian composition factors contain some rich subclass? I would like to know some well known subclass of the following group class: \mathcal{G}=class of all groups with all composition factors are non abelian. Obviously \mathcal{G} contains the direct(... • 564 5 votes 1 answer 124 views Silly cohomology computation I am trying to classify the extensions of \mathbb{Z}/2 by \mathbb{Z}, that is, all G that fits in the exact sequence$$0\to \mathbb{Z}\to G\to \mathbb{Z}/2\to 0.$$These extensions (up to ... • 536 0 votes 2 answers 124 views G is an extension of \mathbb{Z}^2 by \mathbb{Z} Show that G=\mathbb{Z}^3 is an extension of \mathbb{Z}^2 by \mathbb{Z}, where the binary operation of G is$$(x_1,y_1,z_1).(x_2,y_2,z_3)=(x_1+x_2,y_1+y_2,z_1+z_2+y_1x_2)$$Is this extension ... • 2,479 0 votes 0 answers 32 views Non-splitting extensions of groups with trivial center In my question I will use the notation of the lecture notes of my commutative group theory lecture (they are in German). In the section about group extensions I am at the point of an extension of a ... 0 votes 0 answers 47 views Relationship between orders of Ng\in \frac{G}{N} and q\in Q if G=N\cdot Q. Let G=N\cdot Q be a finite extension. What is the relationship between the orders of a coset Ng\in \frac{G}{N} and an element q\in Q? I know that by the natural homomorphism$$f:G\longrightarrow ...
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Let $Q$ and $K$ be finitely presented groups with $H^2(Q)$ finitely generated and $H_1(K) = H_2(K) =0$ and $Z(K) \ne 0$ but fg. If we always use the trivial outer action, is it possible to have an ...
Let $\pi,V$ be a representation of a perfect group $G$. I'm interested in sufficient conditions for a semi direct product like $V \rtimes_\pi G$ to be perfect. Requiring that $\pi$ is faithful ...