Tagged Questions

1
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0answers
42 views

Explanation of notations

I was reading Binary icosahedral group in Wikipedia. The author uses $<2,3,5>$ and $(2,3,5)$ to denote the groups. And the Coxeter group of type $H_4$ is denoted by $[3,3,5]$. Could anyone ...
3
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1answer
35 views

Sufficient condition for a direct limit of abelian groups to be infinitely generated

I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
3
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1answer
75 views

Order of the first cohomology group and subgroups

Let $M$ be a $G$-module and $H$ a subgroup of $G$. Is $\# H^{1}(H, M) < \# H^{1}(G, M)$?
3
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0answers
65 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
3
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0answers
135 views

short exact sequence - split, as a semidirect product, with some cohomology

I've looked at several s.e.s. examples and I feel I am quite close but here is a question I am still a little confused on. Let $E$ be a group and $A$ an abelian normal subgroup s.t. have an exact ...
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1answer
143 views

Simple example of a short exact sequence of groups

I am new to this and would like to understand $$0 \overset{a0}{\to} B \overset{a1}{\to} A \overset{a2}{\to} A/B \overset{a3}{\to} 0, $$ where $B \subset A$ and they are both Abelian groups. Also maybe ...
3
votes
0answers
143 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
1
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2answers
134 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
0
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1answer
206 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
5
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1answer
223 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 ...