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In this paper, we study the cohomology of the Morava stabilizer algebra$S(3)$. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm1 \,\, mod \,p $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $\zeta_n\gamma_s$is a nontrivial product in $\pi_*(S)$ by Adams-Novikov spectral sequence, where$\zeta_n$ is created by R. Cohen \cite{Co}, $\gamma_s$ is a third periodichomotopy elements.

The composition of R.~Cohen's elements and the third periodic elements in stable homotopy groups of spheres