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Let $G$ be a finite almost simple group of Lie type acting faithfully andprimitively on a set $\Omega$. We prove an analogue of the Boston--Shalevconjecture for conjugacy classes: the proportion of conjugacy classes of $G$consisting of derangements is bounded away from zero. This answers a questionof Guralnick and Zalesski. The proof is based on results on the anatomy ofpalindromic polynomials over finite fields (with either reflective symmetry orconjugate-reflective symmetry).

Conjugacy classes of derangements in finite groups of Lie type