Order set-determined group

This group property is about the given group being determined, upto isomorphism, by certain statistic(s)

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 $G$.

Relation with other properties

Stronger properties

• Order statistic-determined group