Global Cayley maps and conjugacy class sizes of maximal unipotent subgroups of finite simple groups of Lie type.
Let \(p\) be a prime and \(G\) be a simple affine algebraic group defined over an algebraic closure \(\overline{\mathbb F}_p\) of the finite field \(\mathbb F_p\). We may assume that \(G\) is a closed subgroup of \(\text{GL}_d(\overline{\mathbb F}_p)\) for some \(d\). Given a power \(q_0\) of \(p\), define \(\sigma_{q_0}(x_{ij})=(x_{ij}^{q_0})\). Then \(\sigma_{q_0}\) is an endomorphism of \(\text{GL}_d(\overline{\mathbb F}_p)\). An endomorphism \(F\) of \(G\) is called a Frobenius endomorphism if there exists an integer \(m\) such that \(F^m(x)=\sigma_{q_0}(x)\) for all \(x\in G\). Define \(q=q_0^{1/m}\) as the level of \(F\). Let \(G_F\) be the group of elements of \(G\) fixed by \(F\). \(G_F\) is as a rule a finite simple group of Lie type. Let \(U_F\) be a \(p\)-Sylow (maximal unipotent) subgroup of \(G_F\). As is well-known, \(|U_F|\) is a power of \(q\) when \(p\) is large enough. The first author [Commun. Algebra 27, No. 5, 2511-2519 (1999; Zbl 0923.20040)] proved that any good prime is ``large enough'' when \(G\) is a classical group or of type \(G_2\). This led to the conjecture that \(|U_F|\) is a power of the level \(q\) of \(F\) when \(p\) is good without further restrictions on \(G\). In the present paper, the authors prove this conjecture. If \(p\) is a bad prime for \(G\), then the conjecture will be false in general. In the last part of the paper, the authors investigate the existence of power series inducing isomorphisms (global Cayley maps) between \(U_F\) and its Lie algebra.
scientific article; zbMATH DE number 5999962
scientific article