All Questions
Tagged with finite-groups group-cohomology
101 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
26
votes
2
answers
1k
views
Is the cohomology ring of a finite group computable?
Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
- 3,230
24
votes
0
answers
813
views
Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
- 1,433
19
votes
1
answer
512
views
Is the map $\mathrm H^4(S_{24}) \to \mathrm H^4(M_{24})$ surjective?
The group $S_{24}$ of permutations of $24$ things has fourth integral cohomology $\mathrm H^4(S_{24};\mathbb Z) \cong \mathbb Z/2 \oplus \mathbb Z/2 \oplus \mathbb Z/12$. According to Sikiric and ...
- 54.6k
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
17
votes
1
answer
925
views
Crossed modules and degree-3 group cohomology
It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, ...
- 1,579
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
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, ...
- 12.8k
16
votes
1
answer
1k
views
Cohomology of central extensions of groups
Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the ...
- 1,579
15
votes
5
answers
3k
views
Tate Cohomology via stable categories
Situation
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by $\underline{G\text{-mod}}...
- 2,756
15
votes
1
answer
461
views
H_3 of SL(n,Z) and SL(n,F_p)
Can anyone tell me what $H_3(SL_n(\mathbb{Z});\mathbb{Z})$ and $H_3(SL_n(\mathbb{F}_p);\mathbb{Z})$ are? It is easy to find references for $H_1$ and $H_2$, but it turns out that I need $H_3$ as well. ...
- 151
14
votes
2
answers
927
views
3-cocycle representatives for the dihedral group $D_{2n}$?
I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...
- 5,425
14
votes
1
answer
699
views
Obstruction to extension of non-abelian groups - finite example?
Let $G$ be a non-abelian group, let $\Pi$ be a group, and let $\eta: \Pi\rightarrow Out(G)$ be a homomorphism, where $Out(G)$ is the group of automorphisms of G modulo the normal subgroup of inner ...
- 283
14
votes
2
answers
6k
views
Group cohomology of the cyclic group
It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...
- 183
14
votes
0
answers
347
views
What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?
Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
- 54.6k
13
votes
2
answers
795
views
Which finite groups have low-degree essential cohomology?
Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...
- 54.6k
13
votes
2
answers
737
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 let'...
- 28.1k
13
votes
0
answers
586
views
Finite groups inside an infinite group with the same homology
Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:
$G$ and $H$ are finite groups and $K$ is an infinite group.
there exist two monomorphisms $G \rightarrow K \leftarrow H$...
- 1,974
12
votes
4
answers
610
views
The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$
Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible 2$...
- 121
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
10
votes
0
answers
134
views
Is there a strictly coassociative resolution of polynomial growth, for a finite group?
Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
- 16.2k
9
votes
2
answers
167
views
Coboundary matrix of bar resolution for group cohomology: do the elementary divisors always divide $|G|$?
Consider the coboundary matrix $C^1(G, \mathbb{Z}) \to C^2(G, \mathbb{Z})$ of the normalized bar resolution of $G$ with coefficients in the trivial $\mathbb{Z}G$-module $\mathbb{Z}$. That is, thinking ...
- 3,230
9
votes
1
answer
804
views
Known results in the Cohomology of finite groups
I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...
- 529
9
votes
1
answer
308
views
How small can the support of a nontrivial $\mathbb F_p$-cocycle on $C_p$ be?
Let $p$ be a prime, and let $\phi : C_p^n \to \mathbb F_p$ be an $\mathbb F_p$-valued $n$-cocycle on $C_p$ (the cyclic group of order $p$) which is not an $n$-coboundary, i.e. $\phi$ represents a ...
- 63.9k
9
votes
1
answer
1k
views
Cohomology of orthogonal and symplectic groups
Hello,
in their book Cohomology of Finite Groups Adem and Milgram investigate the cohomology of the finite orthogonal and symplectic groups only in case $\mathbb{F}_2$.
Let $p$ be a prime dividing ...
- 435
8
votes
3
answers
741
views
Computations in modular cohomology of finite groups
Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...
- 768
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
8
votes
0
answers
128
views
What are the stable cohomology classes of the "orthogonal groups" of finite abelian groups?
Let $A$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $\langle,\rangle : A \times A \to \mathrm{U}(1)$. Then you can reasonably talk about the "orthogonal ...
- 54.6k
8
votes
0
answers
238
views
Is there a finite group with nontrivial $H^2$ but vanishing $H^4$, $H^5$, and $H^6$?
Is there a finite group $G$ such that the group cohomology $\mathrm{H}^2_{\mathrm{gp}}(G; \mathbb{Z}/2)$ is nontrivial but $\mathrm{H}^4_{\mathrm{gp}}(G; \mathbb{Z}/2)$, $\mathrm{H}^5_{\mathrm{gp}}(G;...
- 54.6k
7
votes
2
answers
663
views
Cohomology of $\operatorname{GL}_3(\mathbb{F}_2)$
I’m wondering what is known about the cohomology of $\operatorname{GL}_3(\mathbb{Z}_2)$, the general linear group of $3\times3$ matrices over the finite field $\mathbb{Z}_2$. There are results in ...
- 545
7
votes
2
answers
494
views
How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?
Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
- 54.6k
7
votes
3
answers
1k
views
n-cocycles of finite abelian groups from cohomology group
Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$.
(1) What is the explicit forms of its cohomology group (see my definition) in a ...
- 10.5k
7
votes
2
answers
701
views
Common Computations in Group Cohomology
Let G=A⋊B, where A and B are abelian, and of coprime order. It seems, from my computations (and correct me if I'm wrong), that Z1(Cp,Cq) is trivial, for p and q different primes. Meaning that ...
- 71
7
votes
1
answer
376
views
When is an almost simple group a split extension of its socle?
Here an almost simple group is a finite group whose socle (product of all minimal normal subgroups) is a nonabelian simple group. As an extension of its socle, an almost simple group could be split or ...
- 767
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
224
views
How many values of a group cocycle are required to know its cohomology class?
Suppose I have a finite group $G$, and a group cocycle $\varphi\in Z^n(G,U(1))$ (trivial action on $U(1)$) evaluated at $k$ distinct values of the inputs $g_1,...,g_n$. That is, I am given a set of ...
- 219
7
votes
0
answers
245
views
Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$
$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...
- 6,777
7
votes
0
answers
141
views
Small modules over finite group with large cohomology
Looking at this Example of group cohomology not annihilated by exponent of $G$? I stumbled upon one question I couldn't solve (probably because it's hard), so I post it here.
Using Lyndon resolvent, ...
- 4,600
7
votes
0
answers
229
views
Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$
Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ?
I have looked at the very few books on cohomology of groups ...
- 1,766
7
votes
0
answers
171
views
Can a finite group have its third cohomology determined by a quotient group?
Does there exist a finite group $G$ satisfying either of the following?
$G$ admits a non-trivial, proper, central subgroup $A$ such that every (normalized) 3-cocycle of $G$ is cohomologous to a (...
- 822
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
2
answers
271
views
Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
- 14.2k
6
votes
1
answer
652
views
Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?
I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
user121195
6
votes
2
answers
399
views
A finite group that splits and does not split
Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup)
such that there exists an epimorphism $$\tau \colon ...
- 11.3k
6
votes
1
answer
579
views
first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$
Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard ...
- 1,298
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
1
answer
219
views
Cohomology of finite $p$-groups over integers in local fields
Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).
In ...
- 435
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
312
views
Is group cohomology with the inversion action order two?
Let $\mathbb{Z}^\text{inv}$ denote the $\mathbb{Z}/2$-module defined by the inversion action on $\mathbb{Z}$. Let $G_0$ be a finite group that acts trivially on $\mathbb{Z}^\text{inv}$. Then one can ...
- 1,209
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
2
answers
430
views
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...
- 37k