# Elementary $p$-subgroups of a compact Lie group

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.