7
votes
Accepted
Poincaré duality and Mayer–Vietoris sequence
$\DeclareMathOperator\PD{PD}\DeclareMathOperator\cd{cd}$You need two ingredients which Davis assumes the reader is aware of:
Strebel's theorem: If $G$ is a $\PD(n)$ group, then every infinite index ...
- 10.4k
6
votes
Accepted
Is $\mathbb Z$ prime in the class of abelian groups?
If I understand correctly, the question is equivalent to whether the class of abelian groups with no $\mathbf{Z}$ direct factor is stable under taking direct products.
But this is clearly true, since ...
- 62.3k
4
votes
Accepted
Explicit $2$-cocycle for $2^{1+2n}_+$
When $p=2$, the bilinear form given by commutators no longer specifies the extraspecial group. You need to use a quadratic form $q\colon V \to \mathbb{Z}/2$ giving the square map. A commutator in a ...
- 13.9k
4
votes
Examples for Bogomolov multiplier of finite group
I realize that I'm chiming into a rather old question here, but it turns out that I wanted to know the answer myself awhile ago, and it seems that there's an interesting story lurking here still.
To ...
- 646
3
votes
Smallest $n$ for which $G$ embeds in $S_n$?
There has been some recent progress on algorithms for this problem. Das & Thakkar STOC '24 give the following algorithms:
For groups with no abelian normal subgroups, given by a generating set of ...
- 2,968
1
vote
Block-diagonal embedding of $U(n)$ into $U(mn)$
I claim when $m,n>1$ the subgroup $U(n)\subset U(mn)$ is never normal. The image of $\alpha=\mathrm{diag}(z,1,\dots,1)$ with $|z|=1$ is
$$\mathrm{diag}(z,1^{n-1},z,1^{n-1},\dots,z,1^{n-1}),$$
where ...
- 2,512
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