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