An inequality detecting nilpotency of finite groups

Abstract

In this short note, we provide an inequality that holds in any finite group, only involving the orders of the elements; we prove that equality holds if and only if the group is nilpotent.

As pointed out by Marty Isaacs, there is a mistake in the proof of Theorem 2 (taken from the American Mathematical Monthly) on which we rely. Theorem 2 is still conjecturally true, but reduces our main theorem to a conjecture. We have therefore withdrawn our paper