Rigid cyclic subgroups in Chevalley groups. I
The author studies overgroups of rigid r-subgroups (where r is a prime) of a Chevalley group G defined over a finite field \(F_ q\), where \(q=p^ k\) for some prime p. Rigid r-subgroups R of G are those for which \(C_ G(R)\) is abelian and has a cyclic r-subgroup. The paper gives a characterization of the overgroups of such R in terms of arithmetic and algebraic-group-theoretic properties, and thereby obtains necessary conditions for a quasisimple group to be such an overgroup. Subsequent papers will use the classification of finite simple groups to describe the lattice of overgroups of rigid r-subgroups. Overgroups of maximal tori have been studied by \textit{G. M. Seitz} [Pac. J. Math. 106, 153-244 (1983; Zbl 0522.20031)] by means of root groups. The author obtains more complete results for fields of small characteristic. The nontrivial elements of rigid r-subgroups are regular semisimple elements of the corresponding connected reductive almost simple algebraic groups \(\bar G\) for which \(G=\bar G'_{\sigma}\), where \(\bar G_{\sigma}\) is the finite kernel of the surjective endomorphism \(\sigma\) of G. (Types \(A_ 1(2\) or 3), \({}^ 2A_ 2(4)\), \({}^ 2C_ 2(2)\), \({}^ 2F_ 4(2)\), and \({}^ 2G_ 2(3)\) are excluded.) Rigid r-subgroups are contained in certain maximal regular tori called strongly regular. Work of \textit{T. A. Springer} [Invent. Math. 25, 159-198 (1974; Zbl 0287.20043)] is the point of departure for the study of the structure of root lattices, which gives the basis for several key results that tie together the arithmetic, algebraic-geometric, and finite-group- theoretic properties of rigid r-subgroups R. A representative example of this is Theorem 3.4, which characterizes rigid r-elements (that is, nontrivial elements of such R) as follows. Let \(r\neq p\) and let t be an r-element of G such that \(C_ G(t)\) (misprinted on p. 698 of the paper) has a cyclic \(S_ r\)-subgroup. Then t is a rigid r-element if and only if it is a regular semisimple element of G and r is not a torsion prime. In this case, \(C_ G(t)\) is a maximal torus of G and t is regular. Other results relate rigid r-elements to nondegenerate maximal tori of \(\bar G_{\sigma}\) and to the arithmetic properties of Zsigmondy primes.
connected reductive almost simple algebraic groups
scientific article; zbMATH DE number 4200488