Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Suppose $A$ and $B$ are (possibly isomorphic, possibly non-isomorphic) groups.

A group extension with normal subgroup $A$ and quotient group $B$ is defined as a group $G$ with a specified normal subgroup $N$ having a specified isomorphism to $A$ and a specified isomorphism from the quotient group $G/N$ to $B$.

In some parts of group theory, such a $G$ is termed an extension of $A$ (the subgroup isomorphic to the normal subgroup) by $B$ (the subgroup isomorphic to the quotient group). In some other areas of mathematics, particularly geometric group theory and homology and cohomology theory, $G$ is termed an extension of the quotient by the normal subgroup, so in this case that would be an extension of $B$ by $A$. A choice of terminology that avoids this confusion is "extension with normal subgroup $A$ and quotient group $B$."

A group extension with normal subgroup $A$ and quotient group $B$ can alternatively be thought of as a group $G$ along with a short exact sequence of groups: $$1 \to A \to G \to B \to 1$$ The group extension problem seeks to classify all group extensions with a specified normal subgroup and a specified quotient group.

  • A group extension is said to be split if there is a transversal function which is a homomorphism.
  • A group extension is split iff it is a semidirect product.
  • The study of group extensions has connections with group cohomology.
  • The theory of group extensions is one of the cornerstones of homological algebra.

References:

https://en.wikipedia.org/wiki/Group_extension

https://www.encyclopediaofmath.org/index.php/Extension_of_a_group

http://mathworld.wolfram.com/GroupExtension.html

292 questions
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Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-module, and let $P_3\to P_2\to P_1\to…
Thomas Browning
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What is the relation between semidirect products, extensions, and split extensions?

When I read the textbook about semidirect products and split extensions, I feel like I'm lacking the intuition behind them and their relations. I was wondering if someone could briefly explain such relations to me. Specifically, this is what I'm…
user58289
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3 answers

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned that this is not true, and that there exist…
Robin Goodfellow
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Central extensions versus semidirect products

Consider an extension $E$ of a group $G$ by an abelian group $A$. $$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$ Two special kinds of extensions are: Central Extensions: $A$ is contained in the centre of $E$. Semidirect products:…
Mike F
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Non-Isomorphic Group Extensions

This is a question from a problem set on group cohomology, a subject I've just begun to learn. Let $B$ be a finite group and $A$ be abelian. I am looking for two groups $G_1$ and $G_2$ such that $G_1$ and $G_2$ are isomorphic as groups but…
Alexander Sibelius
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Semidirect product: general automorphism always results in a conjugation

When $G$ is a group, $N$ is a normal subgroup of $G$ and $H$ is another subgroup of $G$ where $ N \cap H = \{1\} $, the normality of $N$ suggests that we can write, for $n_1, n_2 \in N$ and $h_1, h_2 \in H$, $$ n_1 h_1 n_2 h_2 = n_1 h_1 n_2 h_1^{-1}…
user50229
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Classify all groups containing isomorphic copy of $\mathbb{Z}$ of index $2$.

I have the following question: Classify all groups $G$ containing an isomorphic copy of $\mathbb{Z}$ such that the copy has index $2$ in $G$ There are some candidates: $\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}$, and the infinite…
Ergin Süer
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Residually finite extension of a finite group

Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find it either. I say that a group $G$ is an extension…
Alex Doe
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Representations and central extensions of $A_5$

I'm taking a course on representation theory this semester, and the professor assumed we had some knowledge in central extensions. Unfortunately, I'm not very familiar with the topic, and I'm having some trouble really grasping it. I'm not entirely…
beeselmane
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Dot notation for group extensions

I came across the notation $A.B$ in many occassions while reading papers in Group Theory. However, I have not yet found any description of what this notation is supposed to mean. From context I suppose it means that if $G=A.B$, then $G$ is a…
mathma
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Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$

Let $A$ and $B$ be matrices of finite order with integer coefficients. Let $n\in\mathbb{N}$ and let $G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^n$ be the semidirect product, where the action is $\varphi(n)\cdot (m_1,\ldots,m_n)=A^n (m_1,\ldots,m_n)$, and…
Alejandro Tolcachier
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Group extensions of cyclic groups

Let $A$ be an infinite cyclic group and $B$ be a cyclic group of order $n$. Suppose $$0 \to A \to G \to B \to 0$$ is a short exact sequence of abelian groups. What could $G$ be? It is clear enough that $G = \mathbb{Z} \oplus C_m$ works, for all…
Zhen Lin
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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 $G/H$ onto $K$. Let $X$ be a left transversal of $H$…
N00BMaster
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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 finite subgroup with the same cardinality as $G/N$?…
Alex
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Is $G$ a semidirect product of $Z(G)$ and $\operatorname{Inn}(G)$?

The title pretty much sums it up. $\operatorname{Inn}(G)$ is the group of inner automorphisms, $Z(G)$ is the center. I know that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$. This means that we have a central extension (exact sequence): $$1\to…
geodude
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