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Results tagged with group-cohomology
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user 124004
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
3
votes
Accepted
Cohomological variety in case that Sylow subgroup is elementary abelian
Yes. There is a stronger result too, which predates Quillen's theorem: if $G$ is a finite group whose Sylow $p$-subgroup $P$ is abelian, then the restriction map $H^*(G;\mathbb{F}_p)\rightarrow H^*(P …
- 3,451
10
votes
Accepted
Finite domination and Poincaré duality spaces
Corollary 5.4.2 of Wall's article `Poincaré complexes I', Ann. Math. 86 (1967) 213-245 gives examples of 4-dimensional Poincaré complexes $X$ with fundamental group of prime order $p\geq 23$ for which …
- 3,451
11
votes
Accepted
Non-finitely presented FP groups with cohomological dimension $2$
The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. …
- 3,451
2
votes
Kan–Thurston theorem and R-completion
The Kan-Thurston construction depends not just on the homotopy type of $X$, but it depends very heavily on the exact choice of cell structure for $X$. However, it does have some naturality properties …
- 3,451
1
vote
Which groups have undetectable third U(1)-cohomology?
Inspired by YCor's remark I realized that every finite abelian $p$-group $G$ of rank three will have essential elements in $H^3(G;U(1))$. I think that this works for $p=2$ as well, but here's an argu …
- 3,451
6
votes
Is there a finitely presented group with infinite homology over $\mathbb{Q}$?
This answers the other part of your question, not answered by Thompson's group. For each $i\geq 3$ there is a finitely presented group $G_i$ with the property that $H_i(G_i\mathbb{Q})$ is infinite di …
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4
votes
Accepted
Higher cohomology for trivial module for finite groups of Lie type
Lots is known about the cross-characteristic case: it is the same characteristic case that is the difficult one. The method used was introduced by Quillen, who worked out the full answer for $GL_n$. …
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9
votes
How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?
There is a short exact sequence of coefficients $\mathbb{Z}\rightarrow \mathbb{R}\rightarrow \mathbb{R}/\mathbb{Z}=S^1$. This gives you a long exact sequence of cohomology groups. For a finite group …
- 3,451
8
votes
Cohomology of simple finite groups remembers the group?
This is a remark rather than an answer to your question. If you remove the word `simple' it is easy to find such pairs of finite groups. The first examples I learned were (I think) constructed by At …
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6
votes
Accepted
Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence
The action of the quotient on the cohomology groups of the normal subgroup is the trivial action, because the normal subgroup is central. (Think of group cohomology as a functor of the group: the con …
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1
vote
On the group homology
Groups with $H_i(G;\mathbb{Z})=0$ for all $i>0$ are called acyclic groups. There are lots of them. As Mariano Suarez-Alvarez says, any perfect locally-free group is acyclic. There are also finitely …
- 3,451
8
votes
(co)homology of symmetric groups
Here are some comments, including an answer to (3). Firstly, if you want an actual explicit computation of the mod-2 cohomology of symmetric groups $S_n$ for as large an $n$ as possible, you should l …
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