Order set-determined group

Definition
A finite group $$G$$ is said to be order set-determined if given any other group $$H$$ such that:


 * $$G$$ and $$H$$ have the same order (i.e. total cardinality)
 * The set of orders of elements of $$G$$ is the same as the set of orders of elements of $$H$$

then $$H$$ must be isomorphic to $$G$$.

Stronger properties

 * Order statistic-determined group