Abstract: By combining the complemented and abelian conditions of Sylow subgroups, and utilizing  the supersolvable property of traces of maximal subgroups, we studied the question of whether G was a solvable group (if each maximal subgroup of G had a supersolvable trace), and obtained some sufficient and necessary conditions for solvable groups.

Key words: maximal subgroup, trace, Sylow subgroup, solvable group