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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8
votes
1answer
546 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 ...
90
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 ...
37
votes
3answers
982 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 $k[G]...
14
votes
4answers
859 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
1k 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 ...
16
votes
1answer
736 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 ...
7
votes
2answers
217 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 $\sigma(\tau(a))=(\sigma\...
2
votes
1answer
80 views

Completion and algebraic closure commutable

The following corollary of KrasnerĀ“s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of $\...
6
votes
1answer
235 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 ...
4
votes
2answers
292 views

What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring ...
18
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 ...
11
votes
1answer
474 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 ...
8
votes
2answers
386 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
130 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 $\varepsilon:F_0\to\...
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 F_q}/\...
5
votes
1answer
79 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 :C^n(...
1
vote
1answer
140 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
9
votes
1answer
704 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
328 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
2answers
67 views

When does a representation of $H\subset G$ on $V$ extend to a representation of $G$ on $V$?

Let $G$ be a finite group, $H$ a subgroup, and $\varphi:H\rightarrow GL(V)$ a finite-dimensional representation of $H$ over a characteristic zero, algebraically closed field. Let $\chi$ be the ...
5
votes
2answers
183 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 ...
3
votes
2answers
84 views

Is a group always contained in a group that surjects onto its automorphism group?

Let $G$ be a group. I am interested in embedding $G$ in a group $\tilde G$ such that there is a surjective map $\tilde G\rightarrow\operatorname{Aut}G$ whose restriction to $G$ yields the homomorphism ...
0
votes
1answer
53 views

Cartesian product of compact triangulated spaces

Let $X$ and $Y$ two compact triangulated spaces, I am trying to show that $X\times Y$ is also a compact (this is obvious) triangulated space and $$\chi(X\times Y)=\chi(X)\cdot\chi(Y)$$ Any tips on ...
3
votes
1answer
112 views

First group cohomology and composition factors

Let $G$ be a finite group. Let $k$ be a field ($\text{char}(k)=p>0$). Let $P(k)$ be the projective cover of $k$. Assume that for any nontrivial simple $kG$-module $M$ we have $H^1(G,M)=0$. Does it ...
2
votes
1answer
77 views

Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$.

The present question follows up this one, in which I accidentally asked for less than I actually wanted. Given a group $G$, I would like to find an extension $\tilde G$ of its automorphism group $\...
2
votes
1answer
22 views

first homology group with coefficients in divisible group

I had (perhaps very elementary) doubt in the understanding of the computation of first homology group of a finite group over a divisible group. Let $\pi$ be a finite group of order $n$ and $D$ be a ...
1
vote
2answers
91 views

Show that the categories $G$-mod and $\mathbb{Z}G$-mod are equivalent.

I have another basic question inspired from reading the sixth chapter of Weibel's "An Introduction to Homological Algebra". First version of the question: a bit ambiguous At the first paragraph, ...
0
votes
0answers
40 views

A clarification about the meaning of “Let $\mathbb{Z}$ be *the* trivial $G$-module”.

I have a question regarding a definition/lemma in the book from Charles A. Weibel, "An introduction to Homological Algebra". At page 161, there is a claim starting as follows: Let $A$ be any $G$-...