On a bound of Cocke and Venkataraman

TL;DR

This paper establishes a new upper bound on the order of finite groups based on the number of elements of maximum order, strengthening previous results and classifying groups with fewer than 36 such elements.

Contribution

It introduces a sharper bound on group order involving gcd, prime divisors, and Euler's totient, and classifies groups with fewer than 36 elements of maximum order.

Findings

New upper bound on group order involving gcd and prime divisors

Complete classification of groups with less than 36 elements of maximum order

Unification of partial results related to Thompson's conjecture

Abstract

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of gcd(m,4) and the odd prime divisors of m. We show that |G|\le q(m)k^2/\phi(m) where \phi denotes Euler's totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with k<36. This is motivated by a conjecture of Thompson and unifies several partial results in the literature.