Suzuki-Ree groups as algebraic groups over $\mathbb{F}_{\sqrt{\smash[b]p}}$

Abstract

Among the infinite classes of finite simple groups, the most exotic classes are probably the Suzuki groups and the Ree groups. They are "twisted versions" of groups of Lie type, but they cannot be directly obtained as groups of rational points of a suitable linear algebraic group. We provide a framework in which these groups do arise as groups of rational points of algebraic groups over a "twisted field"; in the finite case, such a twisted field can be interpreted as a "field with $\sqrt{\smash[b]p}$ elements". Our framework at once allows for other, perhaps less known, exotic families of groups. Most notably, there is a class of "mixed groups", introduced by J. Tits but also apparent in the work of Steinberg, and we show that they can be obtained as groups of rational points of algebraic groups over a "mixed field". We show that a base change from $\mathbb{F}_{\sqrt{\smash[b]p}}$ to $\mathbb{F}_p$ transforms twisted groups into mixed groups, and we formulate a notion of "twisted descent" that allows to detect which mixed groups arise in this fashion.