Triviality of products of \(\beta\)-elements in the stable homotopy group of spheres (Q912433)

The \(\beta\)-elements of the title, introduced by \textit{H. Toda} [ibid. 11, 197-251 (1971; Zbl 0228.55015)] and generalized by S. Oka, belong to \(\pi_*(S)\), where S is the sphere spectrum localized at p. The result of this paper about the vanishing of products of such elements for primes \(p\geq 5\), too technical to be stated here in detail, continue earlier work of \textit{S. Oka} and the author [Hiroshima Math. J. 12, 611-626 (1982; Zbl 0515.55010)]. The proofs are based on a study of the self- homotopy groups \([M,M]_*\) of the mod p Moore spectrum M, and make use of earlier results of Oka and the author.