On a family involving R.L. Cohenʼs ζ-element

In 1981, R.L. Cohen constructed an infinite family of homotopy elements ζk∈π⁎(S) represented by in the Adams spectral sequence, where p>2 and k⩾1. In this paper, we show that the composite map ζn−1β1βs+2 is nontrivial in the stable homotopy of spheres π2(p−1)[pn+(s+3)p+(s+2)]−7(S), where p⩾5, n>3 and 0⩽s