All Questions
71 questions
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
4
votes
1
answer
318
views
Can $\text{Aut}(G)$ be extended to contain $G$?
Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence
$$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$
So when $Z\neq 1$, as ...
- 815
3
votes
0
answers
87
views
Stem extensions and quotients of Schur covers
Suppose that $G$ is a finite group, and that $\Gamma$ is a central extension of $G$ by $A$, that is
$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$
with the image of $A$ contained ...
- 1,300
2
votes
0
answers
99
views
Cohomological characterization of when $f: \pi_1(\Sigma_g) \to P$ factors through $F_g$ when $P$ is perfect
In previous questions on this site such as this one, it has been asked when a map $\varphi \colon G \to H$ of finitely generated groups factors through a free quotient meaning that there exists a ...
- 71
2
votes
0
answers
134
views
Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
2
votes
0
answers
91
views
Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms
Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
- 245
5
votes
2
answers
430
views
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...
- 37k
6
votes
2
answers
270
views
Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
- 14.2k
5
votes
1
answer
292
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
- 884
5
votes
1
answer
311
views
Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 813
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
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
1
vote
1
answer
100
views
Cohomological variety in case that Sylow subgroup is elementary abelian
Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V_G$ of ...
- 1,077
3
votes
1
answer
264
views
Extensions of a simple group by an elementary abelian $p$-group
Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other ...
- 385
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
3
votes
1
answer
301
views
The torsion subgroup of the coinvariants for a $G$-module
Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
- 14.2k
2
votes
0
answers
122
views
Different definitions of p-fusion and Mislin's theorem
Currently, I am trying to understand and compute homology of finite groups with coefficients in a field of positive characteristic. So, I was searching for some results that could reduce this problem (...
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
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
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
2
votes
1
answer
356
views
Non-trivial example of $H^2(G,M)$ where $M$ is a non-trivial G-representation
Let $G$ be a finite group; denote by $\mathbb{Z}_2$ the cyclic group of order $2$.
Let $\pi: G \rightarrow \mathbb{Z}_2$ be a non-trivial group homomorphism.
Let M be the $G$ representation $\mathbb{Z}...
- 229
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
2
votes
1
answer
119
views
Find $a$ satisfying $x \cup_1 y = \delta a$ when $x,y \in Z^2(G,\mathbb{Z}_2)$
Let $G$ be a finite group. Let $x,y \in Z^2(G,\mathbb{Z}_2)$ be 2-cocycles. Find $a \in C^2(G,\mathbb{Z}_2)$ such that
\begin{align}
x \cup_1 y = \delta a.
\end{align}
Is there a general solution? Is ...
- 229
0
votes
0
answers
124
views
When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
- 1,215
2
votes
0
answers
118
views
Split extension of finite group and Sylow subgroup by abelian $p$-group
Let $p$ be a prime and let $A$ be an abelian normal $p$-subgroup of a finite group $G$. Hence, for any Sylow $p$-subgroup $H$ of $G$, it holds that $A$ is contained in $H$ and so $A$ is a normal ...
- 279
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
4
votes
2
answers
436
views
Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
When $q$ is a power of some odd prime, is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?
A Lie algebra is a vector space $L$ over a field $K$ on which a product operation $[xy]$ is ...
- 271
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
1
vote
0
answers
268
views
On normalized 2-cocycle
Let $G$ be a group acts trivially on an abelian group $A$. Let
$\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume
that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times
...
- 181
2
votes
0
answers
103
views
Lattices with trivial coinvariants for finite groups
Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...
- 2,160
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
3
votes
1
answer
475
views
Group cohomology of $S_3$ in terms of its Sylow subgroups
I am trying to understand $H^*(S_3, M)$ in terms of it's Sylow $p$ subgroups. From III.10.2 and III.10.3 in Brown we know that
\begin{equation}H^n(G,M) = \bigoplus_p H^n(H,M)^G\end{equation}
where $p$...
- 71
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
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
3
votes
1
answer
254
views
Computation of group homology $H_2 ((\mathbb{Z}/3\mathbb{Z}) \rtimes (\mathbb{Z}/4\mathbb{Z}),\mathbb{Z})$
In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we ...
- 2,971
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
3
votes
0
answers
162
views
a universal module for group cohomology?
I noticed the following funny fact when studying cohomology of finite groups. I explain it in the case of $H^2$ but it generalizes.
Consider a two-cocycle $a\in Z^2(G,A)$ where $A$ is a left G-module....
- 6,123
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
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
4
votes
1
answer
243
views
Second cohomology of the adjoint representation
Let $p$ be a prime and let $M_p$ be the $\mathrm{GL}_2(\mathbb{F}_p)$-module of $2 \times 2$ matrices over $\mathbb{F}_p$ with trace $0$ (the action is by conjugation).
Is it true that for $p$ large ...
- 11.3k
1
vote
0
answers
109
views
Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
- 1,813
2
votes
0
answers
100
views
Alternating bihomomorphism is skew of 2-cocycle - relative situation
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ (i.e. $\Omega(g,g)=1$) arises as the skew $\kappa/\kappa^T$ of a 2-...
- 1,813
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
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
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
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
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
4
votes
0
answers
131
views
When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?
Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$.
By Example of a Schur-nontrivial group with no abelian subgroup ...
2
votes
0
answers
345
views
Ring structure on cohomology of groups
Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
- 441