The odd primary order of the commutator on low rank Lie groups

Abstract

Let G be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime p. After localization at p, there is a space A which "generates" G in a certain sense. Assuming G satisfies a homotopy nilpotency condition relative to p, we show that the Samelson product 1 G , 1 G of the identity of G equals the order of the Samelson product ı, ı of the inclusion ı : A → G. Applying this result, we calculate the orders of 1 G , 1 G for all p-regular Lie groups and give bounds of the orders of 1 G , 1 G for certain quasi-p-regular Lie groups.