The Characterization of Almost Simple Groups PGL(2, p) by Their Element Orders

Abstract

For any group G, π _e (G) denotes the set of orders of its elements. If Ω is a subset of positive integers, h(Ω) stands for the number of isomorphism classes of finite groups G such that π _e (G) = Ω. Let Ω = π _e (PGL(2, p)), where p ≥ 5 is a prime number. In this paper, we prove that h(Ω) ∈ {1, ∞ }.

Key Words:

• Element order
• Prime graph
• Projective special linear group

2000 Mathematics Subject Classification:

• 20D05

Acknowledgments

This project was supported by the National Science Foundation of China (Grant No. 10171074), Jiangsu Natural Science Foundation (Grant No. BK200133). The first author would like to thank the University authority sincerely for the partial support of this article.

Notes

^#Communicated by D. Easdown.