Totally deranged elements of almost simple groups and invariable generating sets

Abstract

By a classical theorem of Jordan, every faithful transitive action of a non-trivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention, especially for faithful primitive actions of almost simple groups. In this paper, we show that an almost simple group can have an element that is a derangement in every faithful primitive action, and we call these elements totally deranged. In fact, we classify the totally deranged elements of all almost simple groups, showing that an almost simple group $G$ contains a totally deranged element only if the socle of $G$ is ${\mathrm{Sp}}_4\left(2^f\right)$ or $P{\Omega }_n^{+}\left(q\right)$ with $n=2^l⩾8$. Using this, we classify the invariable generating sets of a finite simple group $G$ of the form $\{x,x^a\}$ where $x\in G$ and $a\in \mathrm{Aut}\left(G\right)$, answering a question of Garzoni. As a final application, we classify the elements of almost simple groups that are contained in a unique maximal subgroup $H$ in the case where $H$ is not core-free, which complements the recent work of Guralnick and Tracey addressing the case where $H$ is core-free.