Elementary abelian p-subgroups of algebraic groups

Abstract

Let ~ be an algebraically closed field and let G be a finite-dimensional algebraic group over N which is nearly simple, i.e. the connected component of the identity G O is perfect, C6(G °) = Z(G °) and G°/Z(G °) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p ~ char K. Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group total if it is in a toms; otherwise nontoral. For several quasisimple algebraic groups and p = 2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is total. For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p = 3 and G of type E6, E 7 and E 8. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, C). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted. Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. To give an application of our methods, we study extraspecial p-groups in E8(~).