A commutativity criterion for closed subgroups of compact Lie groups.

Abstract

Let Γ \Gamma be a closed subgroup of a compact Lie group G G . If the identity component Γ 0 {\Gamma _0} is commutative, and if the order of Γ / Γ 0 \Gamma /{\Gamma _0} is prime to the order of the Weyl group of G G , then it is shown that Γ \Gamma is commutative. If G G is a classical group this extends a theorem of Burnside on finite linear groups. If G G is exceptional this gives some information on Cayley-Dickson algebras, Jordan algebras and the Cayley protective plane.