On Groups with Given Same-Order Types

Abstract

On any group G, define g ∼ h if g, h ∈ G have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G. In this article we prove that a group of the same-order type {1, n} is nilpotent and of the same-order type {1, m, n} is solvable.

Key Words:

• Nilpotent groups
• Same-order type
• Solvable groups

2000 Mathematics Subject Classification:

• 20D60
• 20D06

ACKNOWLEDGMENT

The author would like to thank Professor M. Isaacs for his guidance to the proof of Lemma 1.7.

Project supported by the NNSF of China (No. 11026195) and the foundation of Educational Department of Hubei Province in China (No. Q20111901) and the foundation of Hubei University for Nationalities (No. MY2011T001).

Notes

Communicated by J. Zhang.