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81
votes
1answer
2k views

Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G\: \text{ s.t. } \:H\cong G/N$?

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 for groups for which this ...
22
votes
2answers
523 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 ...
20
votes
0answers
204 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
527 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
700 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 ...
10
votes
1answer
216 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
738 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
1answer
328 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
548 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
4answers
539 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 ...
9
votes
2answers
591 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
2answers
605 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 ...
8
votes
1answer
152 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 ...
7
votes
2answers
227 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
406 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 ...
7
votes
1answer
82 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
129 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
2answers
307 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 ...
6
votes
5answers
198 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
160 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 ...
6
votes
1answer
345 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 ...
6
votes
1answer
199 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
321 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
141 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
340 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
623 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 ...
6
votes
1answer
178 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
412 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
0answers
129 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
82 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
116 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
87 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
246 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 ...
5
votes
1answer
94 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
59 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
444 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
243 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 \ ...
5
votes
1answer
79 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
93 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
61 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 ...
5
votes
0answers
157 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
0answers
162 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
235 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 ...
4
votes
1answer
96 views

Order of the first cohomology group and subgroups

Let $M$ be a $G$-module and $H$ a subgroup of $G$. Is $\# H^{1}(H, M) < \# H^{1}(G, M)$?
4
votes
1answer
466 views

Group cohomology of finite groups

I wonder if the group cohomology of a finite group $G$ with coefficients in $\mathbb{Z}$ is finite. This statement may be too strong. I am interested in, for instance, dihedral group. $$ ...
4
votes
1answer
471 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. ...
4
votes
1answer
53 views

Recovering a character from a cohomology class

Let $G$ be a finite group, and consider $\mathbf Z, \mathbf Q$ and $\mathbf Q/\mathbf Z$ as trivial $G$-modules. Then $H^1(G, \mathbf Q/\mathbf Z) = {\rm Hom}(G, \mathbf Q/\mathbf Z) = \widehat{G}$. ...
4
votes
1answer
194 views

What is the difference between a divisible group and a uniquely divisible group?

I have been looking a cohomology where it is known that uniquely divisible modules have trivial cohomology. But in the case of $\mathbb{Z}$-modules I know $\mathbb{Q}$ has trivial cohomology since its ...
4
votes
1answer
312 views

Is a projective (etc.) $G$-module also a projective $G'$-module?

I'm learning some group cohomology from the third section in Serre's Local Fields, and I'm up to the section on change of group. If $f:G'\rightarrow G$ is a homomorphism of groups and $A$ is a ...
4
votes
1answer
86 views

find a $k[G]$-module basis for a $k$ module with $G$ action

Given a ring $k$, a finite group $G$ and a free $k$-module $M$ with a free action of $G$, why is $M$ a free module over the group ring $k[G]$? (how do I find a $k[G]$ basis for $M$?)