Exponent Preserving Subgroups of the Finite Simple Groups

TL;DR

This paper characterizes finite simple groups that contain proper subgroups sharing the same exponent, providing explicit examples for each such group, thereby advancing understanding of subgroup structures in finite simple groups.

Contribution

It identifies all finite simple groups with proper subgroups having the same exponent and constructs explicit examples for these cases.

Findings

Finite simple groups with proper subgroups of the same exponent are classified.

Explicit examples of such subgroups are provided for each identified group.

The results deepen understanding of the subgroup structure related to element orders in finite simple groups.

Abstract

Given a group G denote with exp(G) its exponent, which is the least common multiple of the order of its elements. In this paper we solve the problem of finding the finite simple groups having a proper subgroup with the same exponent. For each G with this property we will give an explicit example of H<G with exp(G)=exp(H).