All Questions
27 questions with no upvoted or accepted answers
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
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
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
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
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
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
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
0
answers
330
views
Cocycle condition for 2-groups
I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
- 113
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 ...
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
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
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
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 (...
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
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
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
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
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
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
1
vote
0
answers
263
views
When can a 2-cocycle on a subgroup can be extended?
This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?
I am asking this as a new question as I already asked that user but got no ...
- 265
1
vote
0
answers
207
views
Golod Shafarevich Inequality and Inequalities among higher Cohomology groups
As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...
- 493
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