On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields

The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Al, 2Dl, 2E6, 3D4 in the case when the larger field F is an algebraic extension of the smaller nonperfect field K of exceptional characteristic for the group G(F) (characteristics 2 and 3 for the type 3D4 and only 2 for other types). It turned out that apart from, perhaps, the type 2Dl, such intermediate subgroups are standard, that is, they are exhausted by the groups G(P)H for some intermediate subfield P, K ⊆ P ⊆ F, and of the diagonal subgroup H normalizing the group G(P). In this note, it is established that intermediate subgroups are also standard for the type 2Dl

Integral Representation and the Computation of Multiple Combinatorial Sums from Hall’s Commutator Theory

In this paper we prove a series of combinatorial identities arising from computing the exponents of the commutators in P. Hall’s collection formula. We also compute a sum in closed form that arises from using the collection formula in Chevalley groups for solving B. A. F. Wehrfritz problem on the regularity of their Sylow subgroups

Twisted conjugacy classes in twisted Chevalley groups

Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.