On arithmetical properties and arithmetical characterizations of finite groups
Natalia V. Maslova
TL;DR
This work surveys how arithmetical parameters of finite groups, notably the element-order spectrum $\omega(G)$ and the prime spectrum $\pi(G)$, determine or constrain group structure via the Gruenberg--Kegel graph $\Gamma(G)$. It contrasts characterization by spectrum with recognition by GK graph, detailing criteria, known results, and open problems for simple and almost simple groups, including extensions to automorphic covers and isomorphism-type GK graphs. The paper consolidates major theorems (Mazurov–Shi, Gruenberg–Kegel, Vasil'ev–Gorshkov) and presents probabilistic and combinatorial bounds on the number of groups sharing a GK graph, alongside structural results for connected and disconnected GK graphs and SRG classifications. Overall, it maps the landscape of when a finite group's arithmetical data uniquely or nearly uniquely determine the group, and it highlights both rigorous classifications and rich open questions with broad implications in finite group theory and related combinatorics.
Abstract
Arithmetical properties of a finite group are properties of the group which are defined by its arithmetical parameters such as the order of the group, the element orders and so on. In this paper, we discuss a number of results on arithmetical properties of finite groups and characterizations of finite groups by their arithmetical parameters obtained recently. This paper is based on the talk by the author given in frame of (WM)^2 - World Meeting for Women in Mathematics-2022.