Products of Greek letter elements dug up from the third Morava stabilizer algebra

Abstract

In [Hiroshima Math. J. 12 (1982) 611–626], Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: ${\alpha }_1{\gamma }_t$, ${\beta }_2{\gamma }_t$, $\langle {\alpha }_1,{\alpha }_1,{\beta }_{p∕p}^p\rangle {\gamma }_t{\beta }_1$ and $\langle {\beta }_1,p,{\gamma }_t\rangle$ for $t$ with $p\nmid t\left(t^2-1\right)$ for a prime number $p>5$.