The conjugate classes of Chevalley groups of type ( G2)

Abstract

In this paper we determine the conjugate classes of Chevalley groups of type (G,) over finite fields of characteristic f 2, 3. The definition and properties of such groups are given in Chevalley [I] and Ree [5]. A matrix representation of these groups can be found in [4]. The main tools of our investigation are the properties of Chevalley groups in general ([I] III), those of groups of type (G,) ([5] Sections 2 and 3), a theorem of Lang [3], and Sylow’s theorem. Let G be a Chevalley group over a finite field K. Denote by K* the multiplicative group of K, We shall use the notation x,.(t), vr , fir , U, Js W, UC as defined in [I]. In determining the conjugate classes of G, one has to solve various equations of type g-lxg = y in G. The main tools for solving such equations are the fundamental properties (1.1) and (1.2) of G, according to which elements of G are written and compared, and the conjugation rules (1.3)-(1.6). (1.7)-(1.9) are also useful. These are consequences of Theorem 2 and related lemmas in [I].