Refining invariants of finite groups with class functions
Abstract
Many numerical invariants of a finite group arise as the multiplicity of the trivial character in a naturally associated class function, and the remaining multiplicities carry finer structural information. For the number of conjugacy classes of elements whose order involves only a chosen set of primes, the natural such class function is a generalized character interpolating between the classical conjugating character and the regular character. We prove that it is a character in two important general cases, obtained independently by Robinson, and we rule out a family of groups and characters in which, we argue, a counterexample would be most likely to arise. Furthermore, we characterize when these class functions are compatible with passing to a subgroup and thereby sharpen a theorem of Sangroniz. The analogous class function for real elements is shown to always give a character. Finally, we give a dual construction that, together with the recently proved McKay Conjecture, yields a best-possible criterion for a finite group to have an abelian Sylow p-subgroup.
Related
- Arithmetic invariants for finite simple and related groups
- Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
- Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions
- On finite simple groups possessing a conjugacy class of size 3p^n
- Sphere decomposition for finite subgroups of 3-manifold mapping class groups
BibTeX
@misc{schroeder2026refining,
title = {{Refining invariants of finite groups with class functions}},
author = {Christopher A. Schroeder},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26464},
eprint = {2609.26464},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26464},
}
RIS
TY - UNPB
TI - Refining invariants of finite groups with class functions
AU - Christopher A. Schroeder
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26464
UR - https://arxiv.org/abs/2609.26464
AB - Many numerical invariants of a finite group arise as the multiplicity of the trivial character in a naturally associated class function, and the remaining multiplicities carry finer structural information. For the number of conjugacy classes of elements whose order involves only a chosen set of primes, the natural such class function is a generalized character interpolating between the classical conjugating character and the regular character. We prove that it is a character in two important general cases, obtained independently by Robinson, and we rule out a family of groups and characters in which, we argue, a counterexample would be most likely to arise. Furthermore, we characterize when these class functions are compatible with passing to a subgroup and thereby sharpen a theorem of Sangroniz. The analogous class function for real elements is shown to always give a character. Finally, we give a dual construction that, together with the recently proved McKay Conjecture, yields a best-possible criterion for a finite group to have an abelian Sylow p-subgroup.
ER -