  In this paper we consider two functions related to the arithmetic and
geometric means of element orders of a finite group, showing that certain lower
bounds on such functions strongly affect the group structure. In particular,
for every prime $p$, we prove a sufficient condition for a finite group to be
$p$-nilpotent, that is, a group whose elements of $p'$-order form a normal
subgroup. Moreover, we characterize finite cyclic groups with prescribed number
of prime divisors.
