a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).

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87
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1answer
2k views

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
35
votes
3answers
888 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
23
votes
0answers
337 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
16
votes
2answers
2k views

What is the intuition between 1-cocycles (group cohomology)?

This is, I'm sure, an incredibly naive question, but: is there a simple explanation for why one should be interested in 1-cocycles? Let me explain a bit. Given an action of a group $G$ on another ...
15
votes
1answer
701 views

Group cohomology versus deRham cohomology with twisted coefficients

Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
12
votes
3answers
1k views

Group cohomology intuition

Group cohomology is one thing I don't seem to get my head around. I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be ...
12
votes
1answer
1k views

Interpretations of the first cohomology group

I've been revisiting group cohomology, and I realized that there is something I never quite understood. Let $G$ be a finite group, and let $A$ be a $G$-module (i.e. $\mathbb{Z}[G]$-module). Then the ...
11
votes
4answers
793 views

Why is the cohomology of a $K(G,1)$ group cohomology?

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular ...
10
votes
4answers
817 views

What do higher cohomologies mean concretely (in various cohomology theories)?

Superficially I think I understand the definitions of several cohomologies: (1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't ...
10
votes
2answers
196 views

Which groups act freely on $S^n$?

When $n$ is even, it is easy to classify groups which act freely on $S^n$ using degree theory: if $G$ acts on $S^n$, then associating to each element $g \in G$ the degree of the map obtained from ...
10
votes
1answer
246 views

When is $\mathbb{Z}$ a flat $\mathbb{Z}G$-module?

Suppose that $\mathbb{Z}$ is a flat $\mathbb{Z}G$-module for a group $G$. Question: Is $G$ the trivial group ? Nb. I know that the question can be answered affirmatively if $G$ is finitely ...
10
votes
1answer
449 views

Finite groups with periodic cohomology

I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
9
votes
2answers
702 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
9
votes
1answer
256 views

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 ...
8
votes
1answer
633 views

Calculating the group co-homology of the symmetric group $S_3$ with integer coefficients.

I have been trying for a while to make sense of Ex V.3.5 & Ex III.10.1 in Brown's book 'Co-homology of Groups': Calculate the Co-homology of $S_3$ with co-efficients in $\mathbb{Z}$, possibly ...
8
votes
1answer
494 views

Torsion-free virtually-Z is Z

It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan... Is there a simple cohomological proof of the fact that a torsion-free group ...
8
votes
1answer
116 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n ...
8
votes
1answer
164 views

All possible extensions of $S_3$ by $\mathbb{Z}$?

Write $S_3$ for the symmetric group on 3 letters. The question: What are the possible extensions of $S_3$ by $\mathbb{Z}$ (up to equivalence)? (To avoid ambiguity, by an extension of $G''$ by ...
8
votes
1answer
998 views

Direct sum commuting with homology functor

I'm trying to understand a fact about commutation between homology functors and direct sums. In particular, let $G$ be a group of type $FP$ (i.e. there exists a projective resolution of finite length ...
8
votes
0answers
137 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
7
votes
2answers
378 views

Use of noncommutative group cohomology

I have seen a lot of places where the group cohomology when a group acts on a module, is extensively used. But beyond seeing the definition and some claims of partial results, I havent seen any uses ...
7
votes
2answers
341 views

What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I've seen two "standard" complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis ...
7
votes
2answers
278 views

Applications of group cohomology to algebra

I started learning about group cohomology (of finite groups) from two books: Babakhanian and Hilton&Stammbach. The theory is indeed natural and beautiful, but I could not find many examples to its ...
7
votes
2answers
204 views

Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has ...
7
votes
1answer
318 views

Why is the Herbrand quotient of the dual $\hat{A}$ equal to the inverse of the Herbrand quotient of $A$ in this situation?

I'm reading Serre's Local Fields, and I'm trying to understand the proof of Prop. 9 in $\S$5 of Chap. 8 (p.136). First, the setup: $p$ is a prime number $G$ is a cyclic group of order $p$ $A$ is a ...
7
votes
1answer
120 views

Reference request: Introduction to Finite Group Cohomology

I don't know anything about group cohomology and I'd like to. What is the best text to learn this subject? I'd prefer as soft an introduction as possible - that is, lots of motivation, lots of ...
7
votes
1answer
122 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
7
votes
1answer
146 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
7
votes
1answer
41 views

Augmentation ideal and the abelianization of $G$

On a qual problem recently, I came across the following fact: If $G$ is a finite group, and $\mathfrak{a}$ is the augmentation ideal of the integral group ring $\mathbb{Z}G$, then ...
7
votes
0answers
140 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
7
votes
0answers
382 views

How to think about cup product in group cohomology?

I am reading the corresponding section in Cassels-Frohlich. This is the axiomatic definition of cup products given there: (I'm copying from Alison Miller's answer here) The cup product is a family ...
6
votes
5answers
260 views

(Elementary) applications of group (co-)homology

I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology. My background is, ...
6
votes
2answers
122 views

$H^1(G,A)$ is killed by $|G|$ : Proof on the level of cocycles

Let $G$ be a finite group and $A$ a $G$-Module. It is well-known that $H^q(G,A)$ is killed by $|G|$ for all $q \geq 1$. This is usually proved using Restriction-Corestriction (applied with the trivial ...
6
votes
1answer
229 views

$H^1$ of $\Bbb Z$ as a trivial $G$-module is the abelianization of $G$ [duplicate]

Let $G$ be a group and $\mathbb{Z}$ regarded as a trivial $G$-module. As title, I'm trying to prove that $H^1(G,\mathbb{Z})$ is isomorphic to $G/[G,G]$. It is easy to see that $H^1$ is isomorphic to ...
6
votes
2answers
488 views

Cohomological dimension of a group

What is an intuitive, straightforward explanation of the cohomological dimension of a group ? How does one compute the cohomological dimension of a group ? Is there a good reference that explains ...
6
votes
1answer
573 views

Cohomology of finite cyclic groups

I got stuck on the following: Let $G$ be a finite cyclic group. Then it is a well-known fact, that one can compute its Tate-cohomology groups from the complex $$\cdots\xrightarrow{\;\tau ...
6
votes
1answer
139 views

Is there an interpretation of higher cohomology groups in terms of group extensions?

1) Consider a group $G$ and a $G$-module $A$. Then it is well-known that there is a $1-1$ correspondence between elements of $H^2(G,A),$ and group extensions $1\rightarrow A \rightarrow H\rightarrow ...
6
votes
1answer
150 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
6
votes
1answer
713 views

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 ...
6
votes
1answer
183 views

$\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$

I'm trying to read (part of) "The Presentation Rank of a Direct Product of Finite Groups" / Cossey, Gruenberg, Kovacs (Journal Of Alegebra 28, 597-603 (1974)). Here are some basic assertions I need ...
6
votes
1answer
601 views

Group cohomology and Shapiro's lemma

This is a stupid question about group cohomology, but it confuses me a lot. Basically I think that the problem is the fact that I do not really understand Shapiro's lemma. Say we take a profinite ...
6
votes
1answer
83 views

Why is $H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)$?

As the question title suggests, what is the easiest way to see that there is an isomorphism$$H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)?$$
6
votes
1answer
362 views

The Zig Zag Lemma in Cohomology

I´m reading the Zig Zag lemma in Cohomology and i want to prove the exactness of cohomology sequence at $ H^k(A)$ and $H^k(B)$ : A short exact sequence of cochain complexes $ 0 \to A \ ...
6
votes
0answers
131 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
6
votes
0answers
103 views

What are the homology groups of an abelian group?

What are the homology groups of an abelian group? I know there are simple answers in certain cases (e.g. I believe $H_2(A; \mathbb{Z}) = \wedge^2 A$), but it's surprisingly difficult to find any ...
6
votes
0answers
140 views

Does $f(gh)=gf(h)+f(g)$ make $H^1(G,A)$ small enough?

I'm currently reading the text Beginnings of Group Cohomology, trying to understand $H^1(G,A)$ for $G$ a group and $A$ a $G$-module. In the article, the space $H^1(G,A)$ is motivated as a space of ...
5
votes
2answers
89 views

Showing that $\operatorname {Br}(\Bbb F_q)=0$

I want to prove that $\operatorname {Br}(\Bbb F_q)=0$ using the cohomological description of the Brauer group. We have: $\operatorname {Br}(\Bbb F_q)=H^2(\operatorname {Gal}(\overline {\Bbb ...
5
votes
2answers
127 views

Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
5
votes
1answer
742 views

Exercises in Group Cohomology

I'm interested in finding a textbook to learn group cohomology, a book that contains a lot of examples and also a lot of good exercises to test my understanding. I would appreciate some feedback. ...
5
votes
2answers
176 views

Action of $G/H$ on $H_n(H;M)$

I'm currently studying group cohomology and have trouble with the Hochschild-Serre spectral sequence. My problem is this: Given a short exact sequence of groups $$ 0 \to H \to G \to G/H \to 0$$ how ...