We investigate structural properties of finite groups that are detected by certain
group invariants arising from Dijkgraaf–Witten theory, a topological quantum field
theory, in one space and one time dimension. In this setting, each finite group
determines a family of numerical invariants associated with closed orientable surfaces,
expressed in terms of the degrees of the complex irreducible characters of
.
These invariants can be viewed as natural extensions of the commuting probability
,
which measures the probability that two randomly chosen elements of
commute and has been extensively studied in the literature. By analyzing these
higher-genus analogues, we establish new quantitative criteria relating the
values of these invariants to key structural features of finite groups, such
as commutativity, nilpotency, supersolvability and solvability. Our results
generalize several classical theorems concerning the commuting probability,
thereby linking ideas from finite group theory and topological quantum field
theory.
Keywords
character degree, conjugacy class, invariant, commuting
probability, TQFT
