On the invariants of finite groups arising in a topological quantum field theory
Christopher A. Schroeder, Hung P. Tong-Viet
TL;DR
This work connects finite group structure with topological quantum field theory by introducing genus-$h$ invariants $q_h(G)$ from Dijkgraaf--Witten theory and proving sharp, genus-spanning criteria for abelianness, nilpotency, supersolvability, and solvability based on inequalities against fixed small groups. It extends the classical commuting-probability framework to all genus $h\ge 1$ and develops a $p$-local variant $q_{h,p'}(G)$, a dual invariant $\widetilde{q}_h(G)$ with analogous criteria, and corresponding normal Sylow-subgroup results. The proofs leverage minimal-counterexample arguments, Brauer character theory, and Frobenius-type group analyses, yielding a coherent Galois- and representation-theoretic perspective on when group structure is detected by these DW-inspired invariants. The paper also highlights asymptotic behavior $q_h(G) \to 1/|G'|$ as $h\to\infty$ and connects to zeta-function viewpoints for irreducible degrees, thereby linking finite-group theory with topological and quantum-field-theoretic techniques.
Abstract
In this paper, 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 $G$ determines a family of numerical invariants associated with closed orientable surfaces, expressed in terms of the degrees of the complex irreducible characters of $G$. These invariants can be viewed as natural extensions of the commuting probability $d(G)$, which measures the likelihood that two randomly chosen elements of $G$ 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.