A Relation in the Stable Homotopy Groups of Spheres

Let p >= 7 be an odd prime. Based on the Toda bracket <alpha(1)beta(p-1)(1), alpha(1)beta(1), p, gamma(s)>, the authors show that the relation alpha(1)beta(p-1)(1)h(2,0)gamma(s)= beta(p)/(p-1)gamma(s) holds. As a result, they can obtain alpha(1)beta(p)(1)h(2,0)gamma(s) = 0 is an element of pi(*)(S-0) for 2 <= s <= p - 2, even though alpha(1)h(2,0)gamma(s) and beta(1)alpha(1)h(2,0)gamma(s) are not trivial. They also prove that beta(p-1)(1) alpha(1)h(2,0)gamma(3) is nontrivial in pi(*)(S-0) and conjecture that beta(p-1)(1) alpha(1)h(2,0)gamma(s) is nontrivial in pi(*)(S-0) for 3 <= s <= p - 2. Moreover, it is known that beta(p/p-1)gamma(3) = 0 is an element of Ext(BP*BP)(5,*) (BP*, BP*), but beta(p/p-1)gamma(3) is nontrivial in pi(*)(S-0) and represents the element beta(p-1)(1) alpha(1)h(2,0)gamma(3).