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?
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.
