Conjugacy class size 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 conjugacy class size set-determined if whenever $H$ is a group such that:

• $G$ and $H$ have the same order (i.e. cardinality)
• The set of sizes of conjugacy classes of $G$ is the same as that of $H$

then $H$ and $G$ are isomorphic groups.