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

learn more… | top users | synonyms (1)

85
votes
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 ...
27
votes
2answers
662 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 ...
21
votes
0answers
271 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 ...
15
votes
1answer
603 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 ...
11
votes
3answers
919 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 ...
11
votes
2answers
972 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 ...
11
votes
1answer
941 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 ...
10
votes
4answers
618 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
1answer
228 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
397 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
4answers
634 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 ...
9
votes
2answers
644 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 ...
8
votes
1answer
543 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
159 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
796 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 ...
7
votes
2answers
353 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
272 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
1answer
430 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 ...
7
votes
2answers
203 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
1answer
105 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
137 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{ ...
6
votes
5answers
231 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
1answer
216 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
1answer
271 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 ...
6
votes
2answers
377 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
2answers
173 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 ...
6
votes
1answer
446 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
124 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
180 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
495 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
298 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
138 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 ...
6
votes
0answers
188 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 ...
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
118 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 ...
5
votes
2answers
108 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
624 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
187 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
5
votes
1answer
91 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 ...
5
votes
1answer
565 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 ...
5
votes
1answer
79 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
5
votes
1answer
76 views

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n ...
5
votes
1answer
82 views

Why is a class in $H^2(K/k,\mathbb{G}_m)$ locally trivial almost everywhere for $K/k$ a finite Galois extension of number fields?

If $K/k$ is a finite Galois extension of number fields and $\kappa\in H^2(K/k,\mathbb{G}_m)=H^2(\mathrm{Gal}(K/k),K^\times)$, then for each prime $v$ of $k$, upon choosing a prime $w$ of $K$ above ...
5
votes
0answers
63 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 ...
5
votes
0answers
71 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 ...
5
votes
0answers
114 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
5
votes
0answers
329 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 ...
5
votes
0answers
332 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
5
votes
0answers
178 views

Dual modules and first cohomology

Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...
4
votes
2answers
320 views

Motivation for Schur multipliers

What are Schur multipliers good for? I should probably clarify what I want. Here is an instructive story of how I came to appreciate complex representations and characters of groups. Basically, I ...