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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Homology group $H_1(G;\mathbb{R})$ is a vector space?

I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I ...
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22 views

Prerequisites to study cohomology?

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make ...
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48 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
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32 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
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33 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
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10 views

Hochschild-Serre spectral sequence for not normal subalgebra

I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here ...
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32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
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23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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17 views

let $G$ be a group with $cd(G)=m$ ,let $U$ be a subgroup of $G$ of finite index in $G$ ,show that $cd(U)=m$ .

let $G$ be a group with $cd(G)=m$ and $U$ be a subgroup of $G$ of finite index in $G$. Show that $cd(U)=m$ . $cd(G)$:a group $G$ has cohomological dimension$\leq n $ ,denoted by $cd(G)\leq n $ if ...
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20 views

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$.

if $H^{n+1}(G,A)=0$ for all $G$-module $A$ ,then $H^{k}(G,A)=0$ for all $k>n$ and for all $G$-modules $A$. any hint or idea or references to study will be great,thanks.
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27 views

if $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$.

If $G$ and $H$ be groups with $\mathbb{Z}G \simeq \mathbb{Z}H$ then $\frac{G}{G^{'}}\simeq \frac{H}{H^{'}}$. It will be great if you help me with this. Any hint or guidance will be great. Thanks.
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75 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
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54 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
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44 views

Cohomology calculation or computation?

I have a terminology question: Does one compute the cohomology of a group, or does one calculate it? Is it more common to speak of cohomology calculation or cohomology computation? Thanks for your ...
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55 views

Proof for a theorem on Cohomology by Tate

I am searching for a reference for the proof of the following theorem. Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = ...
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28 views

Group cohomology via resolutions

I have a basic question about the definition of group cohomology. Suppose $\Gamma$ is a discrete group, $R$ a commutative ring and $V$ is an $R\Gamma$ module. Then, the first definition of group ...
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46 views

Definition of module

In the book's, The theory of numbers, S. Iyanaga. Chater I, Cohomology of groups. What is the meaning of "module A"? Thank you all.
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84 views

Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
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75 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 ...
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1answer
70 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...
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71 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 ...
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44 views

Brauer groups, Milnor k theory and group cohomology

Can anyone suggest some basic material for learning connections between Brauer groups, Milnor k theory and group cohomology. I am an undergraduate. So, I find most of the sources available very hard.
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68 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 ...
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31 views

Computing cohomology of finite groups of Lie type

Let $G_{/\mathbf{Z}}$ be a Chevalley-Demazure group scheme, i.e. a split reductive group scheme over $\mathbf{Z}$. Let $\rho:G\to \operatorname{GL}(V_{/\mathbf{Z}})$ be a representation. If $k$ is a ...
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37 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
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47 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)+ ...
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54 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 ...
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2answers
81 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 ...
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1answer
41 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. ...
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54 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 ...
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29 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 ...
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29 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 ...
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57 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 ...
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38 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 ...
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67 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 ...
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25 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 ...
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1answer
35 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 ...
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2answers
204 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 ...
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52 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 ...
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159 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 ...
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97 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 ...
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113 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 ...
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64 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 ...
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47 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 ...
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63 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')) ...
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61 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 ...
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28 views

Two nonequivalent extensions with the same factors

Let $0 \rightarrow M \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} P \rightarrow 0$ and $0 \rightarrow M \overset{\alpha'}{\rightarrow} E' \overset{\beta'}{\rightarrow} P \rightarrow 0$ ...
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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
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26 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 ...
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48 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 ...