On The ζ_n-1β₂γ_s+3-element And ω_nβ₁γ_s+3-element In The Stable Homotopy Groups Of Spheres

In1981,R.L.Cohen constructed an infinite family of homotopy elementsζk∈π*(S)represented by hobk∈ExtA3,2(p-1)(pk+1)(Z/p,Z/pin the Adams spectraI se-quence,where p>2andk≥1.In[20],X.Liu construct a new nontrivial family in the stable homotopy groups of spheres πpnq+2pq+q-3(s)which is of order p and is rep-resented by kohn∈ExtA3,pnq+2pq+q(Z/p,Z/p)in the Adams spectral sequence,where p≥5is an odd prime,,n≥3and q=2(p-1).In this paper,we show that the composite maps ζn-1β2γ+3and ωnβ1γs+3∈πq[pn+(s+3)p2+(s+5)p+(s+2)]-8(S)are nontrivial in the stable homotopy groups of spheres,where p≥7,n>3,O≤s<p-5and q=2(p-1).