All Questions
Tagged with finite-groups group-cohomology
16 questions
33
votes
3
answers
6k
views
(co)homology of symmetric groups
Let $S_n=\{\text{bijections }[n]\to[n]\}$ be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the $\mathbb{Z}$-modules $H_k(S_n;\mathbb{Z})$? Using GAP, we ...
- 1,589
19
votes
0
answers
604
views
How is this group theoretic construct called?
Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...
user6671
15
votes
5
answers
3k
views
Tate Cohomology via stable categories
Situation
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by $\underline{G\text{-mod}}...
- 2,756
12
votes
0
answers
2k
views
Non split extension isomorphic (as a group) to a split extension
$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
- 1,196
5
votes
1
answer
618
views
Non-vanishing of the Tate-Shafarevich kernel in group cohomology
Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$).
We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...
- 14.2k
17
votes
1
answer
575
views
Group cochains invariant under the action of the symmetric group
Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
15
votes
1
answer
461
views
H_3 of SL(n,Z) and SL(n,F_p)
Can anyone tell me what $H_3(SL_n(\mathbb{Z});\mathbb{Z})$ and $H_3(SL_n(\mathbb{F}_p);\mathbb{Z})$ are? It is easy to find references for $H_1$ and $H_2$, but it turns out that I need $H_3$ as well. ...
- 151
8
votes
1
answer
522
views
Trivial group cohomology induces trivial cohomology of subgroups
From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
- 813
7
votes
1
answer
701
views
Classification of $p$-groups of order $p^n$ with rank $n-1$
Hello,
i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form
$ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow (\mathbb{Z}...
- 1,269
7
votes
0
answers
190
views
Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$
$M_{12}\rtimes \mathbb{Z}_2$ is a maximal subgroup of $M_{24}$, where $M_{24}$ and $M_{12}$ are Mathieu groups . Also, it is known that $H^3(M_{24}, U(1)) \cong \mathbb{Z}_{12}$. I want to find the ...
user35360
6
votes
1
answer
1k
views
Concrete formula for Shapiro's Lemma
I wonder if there is a concrete formula to express the isomorphism in the well known Shapiro's Lemma that $H^i(G, \text{CoInd}_{H}^{G}(M)) \simeq H^i(H, M)$, where $H \subset G$ is a subgroup of $G$, $...
- 121
6
votes
1
answer
980
views
Group (Co)Homology of Symmetric Group
The question concerns the group homology or group cohomology of symmetric groups.
The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$.
groupprops....
- 10.5k
6
votes
0
answers
194
views
What is the value of the fourth cohomology class of $\mathrm{Co}_0$ induced by the 24-dimensional representation?
The group $\mathrm{Co}_0$ has a 24-dimensional module. This induces a map $\mathrm H^4(O(24),\mathbb Z) \to \mathrm H^4(\mathrm{Co}_0,\mathbb Z)$. Has this map been computed? Has the right hand side ...
- 54.6k
5
votes
1
answer
384
views
Which groups have undetectable third U(1)-cohomology?
Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map
$$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
- 54.6k
5
votes
1
answer
311
views
Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 813
4
votes
0
answers
76
views
Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
- 14.2k