# Tagged Questions

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### Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is $\mathbb{C}\Gamma$.
Assume that $G_{1}$ and $G_{2}$ are torsion free groups which satisfies this ...

**1**

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**1**answer

143 views

### What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?

Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial ...

**4**

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**1**answer

214 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

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**1**answer

114 views

### A formula for isotropy group $\pi_1(G_a)$

Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...

**3**

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**1**answer

145 views

### A group 3-cocycle, trivial on a pair of generating subgroups?

I'm looking for an example of the following situation:
A group $G$ generated by finite subgroups $H$ and $K$,
a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
the ...

**5**

votes

**1**answer

429 views

### Can group cohomology be interpreted as an obstruction to lifts?

The standard way to view the first and second group cohomologies is this:
The Standard Story
Let $G$ be a group, and let $M$ be a commutative group with a $G$-action. Then the first cohomology has ...

**2**

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**3**answers

368 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

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**0**answers

217 views

### Kunneth formula (finite group cohomology) for non trivial action of group

Is there a Kunneth formula relating $H^i(k[G],A)\otimes_k H^i(k[G],B)$ and $H^i(k[G],A\otimes_k B)$ where $A\otimes_k B$ is given the diagonal $G$ action ?

**7**

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**4**answers

1k views

### Program for computing group cohomology

Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space?
I mainly care about infinite groups.

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**2**answers

665 views

### Group Cohomology of Symmetric Powers

Let $G$ be a finite group and $V$ be an integral representation of $G$, i.e. a free abelian group of finite rank with $G$-action. Now consider the symmetric power $Sym(V)$ of $V$ over $\mathbb{Z}$, ...

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**2**answers

566 views

### Computations in group cohomology

Hello,
Given a finitely presentable group $G$, I'm interested in the cup-product from $H^1$ to $H^2$ with real coefficients. I want to know if this is explicitly computable (with a computer) with a ...

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**2**answers

351 views

### Multiplication tables for H*(G/P)?

Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ ...

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**2**answers

625 views

### Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?

Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness ...

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**2**answers

259 views

### a balanced presentation of a cyclic-by-cyclic group?

Let $p>2$ be a prime, $C_p$ be the additive group of integers mod $p$. Then the multiplicative group $\{1,...,p-1\}$ of units in the field $Z/pZ$ is cyclic of order $p-1$, it acts on $C_p$ by left ...

**0**

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**1**answer

571 views

### Is it useful to consider cohomology of group representations?

In group representation theory, one attempts to explain and classify (some of) the modules over the group ring $k[G]$, for some field $k$. In group cohomology, one develops the machinery of the ...

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**1**answer

387 views

### Hopf's formula and inflation map

Suppose we have two finite groups given by presentations
$G=F/N_1 , H=F/N_2$
where $N_1 \subset N_2$ and $F$ is a free group of finite rank.
The canonical map $\pi: G \rightarrow H $ induces the ...

**4**

votes

**2**answers

1k views

### On the cohomology ring of the Grassmannian

The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$-dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,N-m)$, the poset of all partitions ...

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**1**answer

544 views

### Fox Calculus and Cohomology.

Good day,
Could someone please give a reference about how to use Fox Calculus to compute the cohomology of a $2$-group $G$ with coefficients in a submodule of $\oplus^n F_2[G]$.
Is there a formula ...

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**1**answer

326 views

### local-global principles in group cohomology

Let $G$ be a (profinite) group. It is known that if $H^n(G_p,A) = 0$ for all $p$, $S_p$ the Sylow subgroups of $G$, then $H^n(G,A) = 0$.
Are there other local-global principles for different sets of ...

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**2**answers

598 views

### Rational Group Cohomology

This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into ...