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In this paper, the nontriviality of homotopy elements \beta _1\varrho _s in the p-primary component of stable homotopy groups of spheres is shown by virtue of the Adams spectral sequence and the May spectral sequence, where \varrho _s is represented by b_0g_{0}\widetilde {\gamma }_{s} in the Adams spectral sequence, p is a prime number greater than 5, and 3\leq s

To determine the stable homotopy groups of spheres π*(S) is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a ϖn-related family of homotopy elements, β1ωnγs, in the stable homotopy groups of spheres, where n > 3, 3 ≤ s < p − 2 and the ϖn-element was detected by X. Liu. [ABSTRACT FROM AUTHOR]