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/varphi (m) where varphi 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.
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