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

In 1981, R.L. Cohen constructed an infinite family of homotopy elements ζ k ∈ π ⁎ ( S ) represented by h 0 b k ∈ Ext A 3 , 2 ( p − 1 ) ( p k + 1 + 1 ) ( Z / p , Z / p ) in the Adams spectral sequence, where p > 2 and k ⩾ 1 . In this paper, we show that the composite map ζ n − 1 β 1 β s + 2 is nontrivial in the stable homotopy of spheres π 2 ( p − 1 ) [ p n + ( s + 3 ) p + ( s + 2 ) ] − 7 ( S ) , where p ⩾ 5 , n > 3 and 0 ⩽ s < p − 4 .