Characterization of some solvable groups by the number of average orders of subgroups

Let [Formula: see text] be a finite group and let [Formula: see text], the average order of its elements. Let [Formula: see text] be the set of average orders of all subgroups of [Formula: see text]. We show that if [Formula: see text] is a finite solvable group with at least three distinct prime divisors, then [Formula: see text]. Moreover, we classify solvable non-[Formula: see text]-groups with [Formula: see text]. For groups of order [Formula: see text] we obtain exact values of [Formula: see text] depending on the isomorphism type, ranging from [Formula: see text] to [Formula: see text]. For groups of order [Formula: see text] we prove [Formula: see text]. We also investigate [Formula: see text]-groups and exhibit the structure of finite [Formula: see text]-groups with exactly five distinct average orders of subgroups.