All Questions
71 questions
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 ...
- 1,991
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 ...
- 11.2k
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 ...
CommunityBot
- 1
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 ...
- 12.9k
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
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 ...
CommunityBot
- 1
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
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 ...
CommunityBot
- 1
5
votes
1
answer
146
views
Generalization of a lemma of Livne
Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} \...
- 5,039
2
votes
1
answer
83
views
How to claculate the $T$-stable subgroup of second cohomology group
Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \...
- 5,039
5
votes
2
answers
1k
views
symmetric 2-cocycle / many projective representations
Let $G$ be a finite group, $k$ the field of complex numbers.
Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
$$\sigma(g,h)=\...
- 1,846
5
votes
1
answer
428
views
Centralizers in the universal central extensions of the alternating groups?
For $n \ge 8$ the Schur multiplier $H_2(BA_n, \mathbb{Z})$ (where $A_n$ denotes the alternating group) stabilizes to $\mathbb{Z}_2$, and hence there is a universal central extension $\widetilde{A}_n$ ...
CommunityBot
- 1
4
votes
1
answer
672
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
CommunityBot
- 1
1
vote
0
answers
207
views
Golod Shafarevich Inequality and Inequalities among higher Cohomology groups
As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...
- 493
2
votes
1
answer
319
views
Successive Schur covers
Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?
- 6,357
12
votes
0
answers
2k
views
Non split extension isomorphic (as a group) to a split extension
$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
CommunityBot
- 1
14
votes
2
answers
927
views
3-cocycle representatives for the dihedral group $D_{2n}$?
I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...
- 10.5k
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'...
- 85.6k
4
votes
2
answers
791
views
A question about finite groups.
Let $k$ be a positive integer. Is it true that any finite group $H$ of cardinal $4k+2$ whose center contains an element $h$ of order $2$ is isomorphic to the direct product $H=(\mathbb{Z}/2\mathbb{Z})\...
- 37.4k
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}...
- 4,425
0
votes
1
answer
487
views
Must finite groups with isomorphic commutators and quotients be isomorphic?
Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...
- 3,176