Homology rings of homotopy associative H-spaces

Let X be a homotopy associative mod p H-space for p an odd prime. The homology H*(X;Fp) is an associative ring, but not necessarily commutative. We study conditions when for elements of H*(X;Fp). Under certain conditions imply for l=p−2 or p−1. These methods can be used to prove results about homology commutators that were previously obtained using the adjoint action [H. Hamanaka, S. Hara, A. Kono, Adjoint action of Lie groups on the loop spaces and cohomology of exceptional Lie groups, Transform. Group Theory (1996) 44–50, Korea Adv. Inst. Sci. Tech.; A. Kono, K. Kozima, The adjoint action of a Lie group on the space of loops, J. Math. Soc. Japan 45 (3) (1993) 495–509; A. Kono, J. Lin, O. Nishimura, Characterization of the mod 3 cohomology of E7, Proc. Amer. Math. Soc. 131 (10) (2003) 3289–3295]. We also generalize results of Kane [R. Kane, Torsion in homotopy associative H-spaces, Illinois J. Math. 20 (1976) 476–485] to nonfinite mod p homotopy associative H-spaces.