Dectecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra

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Let p ≥ 7 p ≥ 7 be a prime number. Let S ( 3 ) S(3) denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres π ∗ ( S ) π _* (S) using H ∗ , ∗ ( S ( 3 ) ) H*,* (S(3)) . In this paper, we determine all nontrivial products in π ∗ ( S ) π _* (S) of the Greek letter family elements α s , β s , γ s α _s, β _s, γ _s and Cohen’s elements ζ n ζ _n which are detectable by H ∗ , ∗ ( S ( 3 ) ) H*,* (S(3)) . In particular, we show β 1 γ s ζ n ≠ 0 ∈ π ∗ ( S ) β _1 γ _s ζ _n ≠ 0 ∈ π _*(S) , if n ≡ 2 n ≡ 2 mod 3, s ≢ 0 , ± 1 s ≡ 0, ± 1 mod p p .