Group with at most n pairwise commuting elements of order dividing n

Definition
A group with at most n pairwise commuting elements of order dividing n is a group with the property that for every natural number $$n$$, any subset $$S$$ of $$G$$ satisfying the property that $$g^n$$ is the identity for all $$g \in S$$, and $$gh = hg$$ for all $$g,h \in S$$, has size at most $$n$$.

Stronger properties

 * Weaker than::Group with at most n elements of order dividing n
 * Weaker than::Multiplicative group of a skew field

Weaker properties

 * Stronger than::Group in which every finite abelian subgroup is cyclic