4
$\begingroup$

How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?

Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}_p$, there is an interesting non-toral elementary $p$-subgroup considered by Vistoli in this paper.

How many other cases are known? For example, how about $G=PU_n$ where $n$ is not a prime?

$\endgroup$
10
$\begingroup$

This is a topic with a long history going (at least) back to Borel, Serre, Steinberg and others. The existence of non-toral elementary abelian $p$-subgroups (i.e. subgroups not contained in a maximal torus of $G$) is equivalent to $H_*(G;\mathbb{Z})$ having $p$-torsion. Newer results include R. L. Griess' paper "Elementary abelian $p$-subgroups of algebraic groups", Andersen–Grodal–Møller–Viruel "The classification of $p$-compact groups for $p$ odd" and J. Yu, "Elementary abelian $2$-subgroups of compact Lie groups". These paper also contain a detailed history of the subject.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.