The odd primary order of the commutator on low rank Lie groups (Q1744610)

Let \(G\) be a simply connected, compact simple Lie group and \(p\) an odd prime. After localization at \(p\), there is a co-\(H\)-space \(A\) and a map \(\iota: A \to G_{(p)}\) such that the homology of \(G_{(p)}\) is generated as an exterior algebra by the image of \(\iota\) [\textit{M. Mimura} et al., Publ. Res. Inst. Math. Sci. 13, 627--680 (1977; Zbl 0383.22007)]. The author compares the order of the commutator \(\langle1_G,1_G\rangle : G \wedge G \to G\) (Samelson product) to the order of \(\langle \iota,\iota\rangle\). He proves that these orders have the same \(p\)-exponent provided \(p\) is large relative to the homotopy nilpotency class of \(G\). For each group \(G\) he can then determine the actual exponents of the order of \(\langle1_G,1_G\rangle\) for all \(p\) up to a finite number of exceptions.