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0
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1answer
43 views

Prove that $H^2(G,{\bf C}^\times ) = {\bf Z}_2$

I want to prove this : $$H^2(G,{\bf C}^\times ) = {\bf Z}_2 $$ where $$ G:={\rm Gal}\ ({\bf C}/{\bf R})={\bf Z}_2=\langle\alpha\rangle$$ Step 1 : normalized $2$-cocycle condition is $$ f(g,h)+ ...
4
votes
1answer
49 views

The first cohomology group $H^1(G,\mathbb{Z})$ for $G$ finite

I want to compute the first cohomology group $H^1(G,\mathbb{Z})$ for $G$ finite. Here is what I have got so far: If $G$ has odd order, $G$ has to act on $\mathbb{Z}$ trivially. Then ...
3
votes
2answers
69 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 ...
2
votes
1answer
36 views

Direct limit of subgroups

Let $G$ be a group and $G^i$ a collection of subgroups which form a direct system over a directed set $I$, so $i\leq j \iff \exists\; \varphi^i_j: G^i\to G^j$ where $\varphi^i_j$ is the inclusion map. ...
0
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0answers
41 views

last part of proof of schur zassenhaus theorem.

Theorem states- Let $G$ be a finite group of order $mn$ and $N$ be a normal subgroup of order $n$, then schur zassenhaus states that there exist a complement of $N$ in $G$ of order $m$ and all such ...
0
votes
1answer
28 views

$H^1(G,A)\rightarrow H^1(H,A)$ is onto

Note that $H^1(G,A)\rightarrow H^1(H,A)^{G/H}$ where $A$ is $G$-module and $H$ is a subgroup in $G$ But I suspect that ${\rm Res}\ : \ H^1(G,A)\rightarrow H^1(H,A)$ may be onto since $ f\in C^n ...
1
vote
0answers
26 views

What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map ...
1
vote
1answer
53 views

Confusion in Serre's Local fields book

I read that for right exact functors we consider left derived functors and the resolutions that we consider are projective resolutions... I read that for left exact functors we consider right derived ...
1
vote
0answers
31 views

Rearding notation of (Relatively)Projective/ (Relatively)Injective in Group cohomology

I am reading Group cohomology from Serre's Local Fields. I got confused with the notation he used... We know that : $A$ is Projective module if $Hom_R(A, \_)$ is exact $A$ is Injective module if ...
1
vote
1answer
65 views

Another description for the map $\text{Ext}^1_\mathbb{Z}(A,G)\to H^2(G,A)$

Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology ...
1
vote
0answers
22 views

Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
1
vote
1answer
27 views

Supplement for reading Group cohomology from Serre Local Fields

I am doing a reading course on Group cohomology... I am supposed to start reading Group cohomology part in Serre's Local fields Book ...
2
votes
2answers
198 views

How can I find $\mathbb Z_4$ as an extension of $\mathbb Z_2$ by $\mathbb Z_2$?

Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)={f_0}$ where ...
1
vote
1answer
44 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
1
vote
1answer
83 views

Understanding a proof about splitting of short exact sequences.

I am reading a paper by Keith Conrad about the splitting of exact sequences. I have a few questions about one particular section. This is Theorem 3.3 in this paper ...
2
votes
1answer
107 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
1
vote
1answer
38 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
1
vote
0answers
33 views

Kunneth formula for group homology

I'm trying to prove Kunneth formula for group homology. $$ 0 \to \bigoplus_p H_p(G,M)\otimes H_{n-p}(G',M') \to H_n(G\times G',M \times M') \to \bigoplus_p Tor_1^{\mathbb Z}(H_p(G,M),H_{n-p-1}(G',M')) ...
0
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0answers
55 views

Homology groups of $SL(2,\mathbb Z)$

I am reading Brown's book "Cohomology of Groups" and I can't solve exercise II.7.1.3.: "It's a classical fact that $SL_2(\mathbb Z) \cong \mathbb Z_6 *_{\mathbb Z_2}\mathbb Z_4.$ Use Mayer-Vietoris ...
0
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0answers
27 views

Two nonequivalent extensions with the same factors

Let $0 \rightarrow M \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} P \rightarrow 0$ and $0 \rightarrow M \overset{\alpha'}{\rightarrow} E' \overset{\beta'}{\rightarrow} P \rightarrow 0$ ...
1
vote
1answer
133 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
0
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0answers
24 views

Is group cohomology functorial in the first argument?

Ie, suppose you have a group $G$ acting on a group $A$ (allowing both to be nonabelian), then suppose you have another group $G'$ acting on $A$, and a group homomorphism $G\rightarrow G'$ (maybe we ...
0
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0answers
41 views

How does the first group cohomology classify torsors?

I know this is true, but I'm having some trouble finding any references on this. I'm in particular interested in the nonabelian case. Specifically, let $G$ be a group acting on a group $A$ (both ...
4
votes
1answer
55 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 $H^1(G, \mathbf Q/\mathbf Z) = {\rm Hom}(G, \mathbf Q/\mathbf Z) = \widehat{G}$. ...
3
votes
1answer
57 views

Why do we have that Hom is an exact functor in the situation described below?

We are given a finite $p$-group $G$ and a finite $G$-module $M$ such that $pM=0$ (therefore $M$ is in particular a $\mathbb{F}_p$-vector space). In addition we have an arbitrary $G$-module $N$ which ...
2
votes
1answer
27 views

Group cohomology of mapping-class-group

Let $MCG_g$ be the mapping class group of closed genus $g$ Riemannian surface. What is the group cohomology $H^n(MSG_g,Z)$ for $n=2$ (and other values).
2
votes
0answers
38 views

$\operatorname{Br}(\Bbb Q_{47})$

I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should ...
2
votes
1answer
53 views

Computing $H^\bullet(\Bbb Z/n\Bbb Z)$

This is related to this other question of mine Showing that $\operatorname {Br}(\Bbb F_q)=0$ in which I also got stuck at writing a free resolution. I want to compute the group cohomology ...
5
votes
2answers
87 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 ...
0
votes
0answers
24 views

free resolutions of $\mathbb Z$ in Mod(G)

Lang Algebra, XX.2,3 I'm asked to show that $E_\bullet \cong F_\bullet$ are isomorphic free resolutions of $\mathbb Z$, in Mod(G), where $E_\bullet$ is the standard complex: $E_i$ is the free ...
0
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0answers
21 views

Sum rules for group-cohomology cocycles?

Consider a cocycle ($\omega$) in $H^n(G,U(1))$, where $U(1)$ is the group of unitary matrices over ${\rm C}$ of dimension 1. Thinking of $U(1)$ as a circle in the complex plane, one can multiply and ...
2
votes
1answer
92 views

Second cohomology group of a perfect group

Consider a finite perfect group $G$ and a $G$-module, $U\left(1\right)$, on which $G$ acts trivially. Here $U\left(1\right)$ is the set of $1 \times 1$ unitary matrices over $\mathbb{C}$. Are there ...
2
votes
1answer
96 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
5
votes
1answer
108 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 ...
3
votes
0answers
62 views

Central extensions of elementary abelian p-groups

Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is : How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n ...
1
vote
1answer
38 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
1
vote
0answers
77 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
1
vote
0answers
33 views

Group cohomology of $\mathbb Z^n$.

I know very little about group cohomology but the following came up in something else I was looking at: Let $c_1,c_2,\ldots,c_n \in \mathbb C$ and let $\mathbb Z^n$ act on $\mathbb C$ via the ...
5
votes
1answer
73 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 ...
0
votes
0answers
38 views

Uniqueness of the cohomological functor

This question is from the chapter 'Cohomology of Groups' by Atiyah and Wall in Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group. Theorem 1 on page 95 says that there is a ...
1
vote
1answer
32 views

A G-isomorphism of certain Hom groups

This question is from 'Cohomology of Groups' by Atiyah and Wall, p.95 of Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group and $A={\rm Hom}_{\mathbb Z}(\mathbb Z[G],X)$ where ...
0
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0answers
74 views

inflation restriction exact sequence

Let $G$ be a profinite group and $H$ be a normal closed subgroup of $G$. For every discrete $G$-module $A$ we have the inflation-restriction exact sequence $$0 \to H^1(G/H,A^H) \to H^1(G,A) \to ...
5
votes
0answers
99 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 $H^2(E,\mathbb{Z})$.?
1
vote
2answers
58 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
1
vote
1answer
65 views

Cohomology of finite p-groups

Given a finite abelian $p$-group $A$ acted on by a finite $p$-group $G$. Under the assumption $\operatorname{H}^1(G,A_1)=0$, where $A_1$ is the set of elements of $A$ having order at most $p$, what ...
5
votes
1answer
63 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
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0answers
69 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 ...
1
vote
1answer
84 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
2
votes
1answer
121 views

Why are projective modules cohomologically trivial?

Let $G$ be a finite group, $H\subset G$ a subgroup, $k$ a commutative ring, $M$ a $kG$-module, $n\in\mathbb{Z}$, and $\hat{H}\,^n(H,M)$ the $n$th Tate cohomology group as defined in this question, ...
7
votes
1answer
91 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 ...