On R L Cohen’s –element

Let p be a prime greater than three. In the p–local stable homotopy groups of spheres, R L Cohen constructed the infinite –element n 1 2 2pnC1 2pnC2p 5.S/ of order p . In the stable homotopy group 2pnC1 2pnC2p2 3.V .1// of the Smith– Toda spectrum V .1/ , X Liu constructed an essential element $k for k 3 . Let ˇ s D j0j1ˇ s 2 ŒV .1/;S 2sp2 2s 2p denote the Spanier–Whitehead dual of the generator ˇ s D ˇ i1i0 2 2sp2 2s.V .1// , which defines the ˇ–element ˇs . Let s;k D ˇ s 1$k . In this paper, we show that the composite of R L Cohen’s – element n 1 with s;n is nontrivial, where n> 4 and 1< s < p 1 . As a corollary, s;n is also nontrivial for 1< s < p 1 .