Group with at most n elements of order dividing n

Definition
A group $$G$$ with identity element $$e$$ is termed a group with at most n elements of order dividing n if the following is true for every natural number $$n$$: the number of $$g \in G$$ such that $$g^n = e$$ is at most $$n$$.

Stronger properties

 * Weaker than::Multiplicative group of a field
 * Weaker than::Group with at most n nth roots for any element

Weaker properties

 * Stronger than::Group in which every finite subgroup is cyclic:
 * Stronger than::Group with at most n pairwise commuting elements of order dividing n
 * Stronger than::Group with finitely many elements of order dividing n
 * Stronger than::Group with finitely many conjugacy classes of elements of order dividing n

Facts
For a finite group, we have a theorem that the number of nth roots is a multiple of n for each $$n$$ dividing the order of the group. Thus, for a finite group, this condition would imply that there are exactly $$n$$ elements of order dividing $$n$$. In fact, a finite group satisfying this condition is cyclic.

A more general question is the following: given any finite group $$G$$ and a natural number $$n$$ dividing the order of $$G$$ such that there are exactly $$n$$ elements whose order divides $$n$$, do those $$n$$ elements form a subgroup? This is the Frobenius conjecture on nth roots.