17 research outputs found
On the characters of Sylow -subgroups of finite Chevalley groups for arbitrary primes
We develop in this work a method to parametrize the set of
irreducible characters of a Sylow -subgroup of a finite Chevalley group
which is valid for arbitrary primes , in particular when is a
very bad prime for . As an application, we parametrize
when .Comment: 22 page
On the coadjoint orbits of maximal unipotent subgroups of reductive groups
Let G be a simple algebraic group defined over an algebraically closed field
of characteristic 0 or a good prime for G. Let U be a maximal unipotent
subgroup of G and \u its Lie algebra. We prove the separability of orbit maps
and the connectedness of centralizers for the coadjoint action of U on (certain
quotients of) the dual \u* of \u. This leads to a method to give a
parametrization of the coadjoint orbits in terms of so-called minimal
representatives which form a disjoint union of quasi-affine varieties.
Moreover, we obtain an algorithm to explicitly calculate this parametrization
which has been used for G of rank at most 8, except E8.
When G is defined and split over the field of q elements, for q the power of
a good prime for G, this algorithmic parametrization is used to calculate the
number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q),
\u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these
calculations can be viewed as an extension of the results obtained in our
earlier paper. In each case considered here there is a polynomial h(t) with
integer coefficients such that for every such q we have k(U(q)) = h(q).Comment: 14 pages; v2 23 pages; to appear in Transformation Group
Primitive permutation groups and derangements of prime power order
Let be a transitive permutation group on a finite set of size at least
. By a well known theorem of Fein, Kantor and Schacher, contains a
derangement of prime power order. In this paper, we study the finite primitive
permutation groups with the extremal property that the order of every
derangement is an -power, for some fixed prime . First we show that these
groups are either almost simple or affine, and we determine all the almost
simple groups with this property. We also prove that an affine group has
this property if and only if every two-point stabilizer is an -group. Here
the structure of has been extensively studied in work of Guralnick and
Wiegand on the multiplicative structure of Galois field extensions, and in
later work of Fleischmann, Lempken and Tiep on -semiregular pairs.Comment: 30 pages; to appear in Manuscripta Mat