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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Fibration coming from a group extension

I am trying to solve the following exercise about classifying spaces (5.1.28) in the book "Algebraic K-Theory and its applications" by Rosenberg: Let $$1 \longrightarrow N \longrightarrow G \...
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Help with Exercise 6.9.2 in Weibels "An introduction to Homological Algebra

I am trying to solve the following exercise: Let $0 \longrightarrow B \longrightarrow Y \overset{\rho}{\longrightarrow} X \longrightarrow 1 $ and $0 \longrightarrow A \longrightarrow X \overset{\pi}{\...
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Difference between the concepts of extension A by B and essential extension for modules?

The following is taken from: is a continuation of the following question Background: Definition 1. A short exact sequence of the form $(f,E,g)\equiv 0\to A\xrightarrow{f}E\xrightarrow{g}B\to 0$ is ...
Seth's user avatar
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Finite image property under extensions

Let $L$ be a group. We say a group $H$ only admits finite images of $L$, if for every homomorphism $f\colon L\to H$ the image $f(L)$ is finite. Now, assume that $G$ is a group and $H\subseteq G$ be a ...
user12345's user avatar
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Extensions of simple Lie groups with a topological splitting

This is a follow-up to my question here.* I'm interested in knowing about (finite-dimensional) Lie group extensions, where $G$ is simple and $H$ is connected (and hopefully abelian): $$1\to H\to G'\to ...
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Central Extensions of the Quaternions

Which groups of order 16 can occur as a central extension of $Q_8$, the group of quaternions, by $\mathbb{Z}/2$? I know that the direct product $Q_8\times \mathbb{Z}/2$ and the semidirect product $\...
JeCl's user avatar
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$\text{GL }_2(\mathbb{Z})$ as an HNN extension

This question arises from section $2$, exercise $13.8$ of Bogopolski's Introduction to Group Theory. I managed to show that $\text{GL}_2(\mathbb{Z})\cong D_4 *_{D_2} D_6$. Now, I want to take a random ...
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Classifying central extensions of perfect groups

In Weibel's book (Exercise 6.9.1), it is claimed that central extensions of the form $0\to A\to X\to G\to 1$ with G perfect are classified by $Hom(H_2(G,\mathbb{Z}), A)$. My thoughts so far lead me ...
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Is a central extension of a torus by a torus a torus?

I have a rather simple question. Let $$ 1 \to T^k \to G \to T^l \to 1 $$ be a central extension of Lie groups. Is it true that $G \cong T^{k+l}$? If not what can it be and what are some simple ...
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Central extension of a Lie group by the circle group $\mathbb{T}$

In page 47 of the book "Loop groups", by Pressley and Segal, we have the following proposition: In the construction, they define a map $C:LX \longrightarrow \mathbb{T}$, from the space of ...
Math learner's user avatar
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Understanding the construction of the split extension of group

This is the 2nd part of another question, mainly general extension. Please have a look to understand the notation. A brief description was copied from that thread, Let $\phi$ be an isomorphism of $G/...
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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-...
Math learner's user avatar
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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 ...
Keshav Srinivasan's user avatar
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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 ...
Rajesh Dey's user avatar
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Splitting of an extension

Let $G$ be a group which is the extension of a free abelian group $A$ of finite rank by a finite simple group $S$. Does $G$ splits over $A$? (that is, $G=F\ltimes A$ for some finite subgroup $F\simeq ...
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Abelian group as second cohomology group of a pair G,M

I'm currently studying group cohomology and in particular group extension; I'm trying to figure out a solution to the following problem: let A an abelian group, is possible to find a group G and a G-...
Lorenzo Ferraiuolo's user avatar
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For a coupling there does not exist an extension.

Let $N$, $Q$ be groups and there is group homomorphism $$\Phi: Q \to Out(N)= Aut(N)/Inn(N),$$ we called such $\Phi$ to be coupling. For every extension of $N$ by $Q$ there exist a coupling. What ...
Devkaran Singh's user avatar
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What is the general notion of central charge?

In the case of a matrix Lie group and a projective representation, central charge is defined here as follows. If $M$ is a matrix Lie group, then elements $G$ of its Lie algebra $\mathbf{m}$ can be ...
mma's user avatar
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Different approaches to classification of finite groups

Currently, I'm approaching the problem of finite group classification, in particular I'm studying group cohomology. I know that to achieve this result one path mathematicians are actually working on ...
Lorenzo Ferraiuolo's user avatar
3 votes
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Best books on finite group extensions

I'm trying to get a deeper understanding of the classification problem for finite groups and in particular the extension problem. I'm looking for some advanced books on the theory of finite group ...
Lorenzo Ferraiuolo's user avatar
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Groups extensions by linear cocycles

Let $0 \to A \to X \to C \to 0$ be an abelian group extension (we require $X$ to be abelian too). Then the group operation on $X$ is described by a $2$-cocycle $c(x, y) = s(x) + s(y) -s(x+y)$ where $s:...
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What is $H^2(U(1),\mathbb Z_2)$?

Let $U(1)$ denote the multiplicative group of complex numbers of modulus $1$ and $\mathbb Z_2:=\mathbb Z/(2\mathbb Z)=\left(\{0,1\},+\right)$ I know that the central group extension $$\mathbb Z_2\...
mma's user avatar
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Property preserved by tensor product/ epimorphic image/ extension [closed]

Let $G$ be a group. Consider $\gamma_1(G):=G$ and $\gamma_{i+1}(G):=[\gamma_i,G]$, we denote $G_{\text{ab}}$ the abelianization of $G$. Then $\gamma_{i+1}/\gamma_{i+2}(G)$ is an epimorphic image of $...
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Cocycles and nontrivial central extensions

Let $G$ be a group and $A$ an Abelian group. In group cohomology, it's a well-known fact that given two cocycles $\omega_1, \omega_2:G\times G\to A$, the central group extensions $G\times_{\omega_i} A$...
mma's user avatar
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Permutations of composition factors of a finite group of finite length.

Let $G$ be a finite group of length $\ell(G)=n$, if $H$ is a decomposition series for $G$, we denote by $L(H)$ the ordered set of its factors. I'm trying to figure out if the set $X_G:=\{\sigma\in\...
Lorenzo Ferraiuolo's user avatar
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Why isn't a coxeter group a HNN-extension?

A doctorate told me to think about why there is no mapping from coxeter groups to $\mathbb{Z}$. This makes sense since HNN-extensions are of the form $$A\star_{\{(\varphi_1 , C,\varphi_2 )\}}=\langle ...
shekh's user avatar
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1 answer
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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 ...
Arthur Filippi's user avatar
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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 ...
Andrea Antinucci's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
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Does the thick subcategory generated by a torsion-free module contain a free module?

Let $R$ be a commutative ring. An $R$-module $M$ is torsion-free, if for every regular (=non-zero-divisor) element $r\in R$ and every $m\in M$ it holds that $rm \neq 0$. We say a full subcategory $\...
Jonas Linssen's user avatar
2 votes
1 answer
103 views

Explicit expression for Bockstein homomorphism in singular cohomology

Let $X$ be a simplicial topological space, with a fixed simplicial decomposition whose vertices are denoted by $i,j,k,...$. For any abelian group $A$ we can consider the singular cohomology groups $H^...
Andrea Antinucci's user avatar
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1 answer
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cochains valued in a central extension in terms of the cochains valued in the subgroup and the quotient.

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\...
Andrea Antinucci's user avatar
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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 ...
HitMan01's user avatar
-1 votes
1 answer
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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 ...
anoniem's user avatar
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8 votes
1 answer
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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 ...
Alex's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
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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 ...
Ian Gershon Teixeira's user avatar
1 vote
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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(...
Infinity_hunter's user avatar
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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 = \...
Lucas's user avatar
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3 votes
1 answer
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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 ...
Cadenza's user avatar
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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(...
Jins's user avatar
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5 votes
1 answer
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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 ...
Ja_1941's user avatar
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2 answers
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$G$ 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 ...
Dima's user avatar
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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 ...
Tim Wittenberg's user avatar
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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 ...
Isaac 's user avatar
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1 answer
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Inverse Sequence of Group Extensions?

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 ...
Jeffrey Rolland's user avatar
6 votes
1 answer
128 views

Perfect semi direct products

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 ...
Ian Gershon Teixeira's user avatar

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