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The relation between elementary abelian  p -subgroups of a connected compact Lie group  G  and the existence of  p -torsion in      H ∗  ( G ) {H^ {\ast } }(G)       has been known for some time [ B-S ]. In this paper we prove that if  G  is any compact Lie group then      H ∗  ( B G ) {H^ {\ast } }(BG)       contains  p -torsion of infinite height iff  G  contains an elementary abelian  p -group not contained in a maximal torus. The hard direction is proven using the double coset theorem for the transfer. A third equivalent condition is also given.

𝑝-subgroups of compact Lie groups and torsion of infinite height in 𝐻*(𝐵𝐺)