On quantitative structure of small Ree groups

TL;DR

This paper investigates the unique structural properties of small Ree groups, demonstrating they are characterized by their order and element order counts, and resolves a problem posed by J. G. Thompson for these groups.

Contribution

It proves small Ree groups are uniquely determined by their order and element order distribution, addressing Thompson's problem for these simple groups.

Findings

Small Ree groups are uniquely identified by their order and element order counts.

The paper confirms Thompson's problem has a positive solution for small Ree groups.

Provides a new characterization method for simple groups based on quantitative structure.

Abstract

The main aim of this article is to study quantitative structure of small Ree Groups $^{2}G_{2}(q)$. Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order. As a consequence, we give a positive answer to J. G. Thompson's problem for simple groups $^{2}G_{2}(q)$.