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.

中文翻译：

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在 Cocke 和 Venkataraman 的边界上

设G是一个有限群，恰好有k个最大可能阶数m 的元素。令q ( m ) 是\(\gcd (m,4)\)和m的奇素数除数的乘积。我们证明\(|G|\le q(m)k^2/\varphi (m)\)其中\(\varphi \)表示欧拉的整体函数。这加强了 Cocke 和 Venkataraman 最近的结果。作为一个应用，我们用\(k<36\)对所有有限群进行分类。这是受 Thompson 猜想的启发，并统一了文献中的几个部分结果。