John R. Britnell

Cyclic, Separable and Semisimple Matrices in the Special Linear Groups Over a Finite Field

A matrix A with minimum polynomial mA and characteristic polynomial cA is said to be cyclic if mA = cA, semisimple if mA has no repeated factors, and separable if it is both cyclic and semisimple. For any set T of matrices, we write CT for the proportion of cyclic matrices in T , SST for the proportion of semisimple matrices, and ST for the proportion of separable matrices. We will write CGL(∞,q) for limd→∞ CGL(d,q), and so on. Wall [6] has shown using generating functions that

On types and classes of commuting matrices over finite fields

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes which are represented in the centralizer of every nilpotent matrix are classified--this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Ko\v{s}ir and Oblak.

The probability that a pair of elements of a finite group are conjugate

Let

Cyclic, separable and semisimple transformations in the special unitary groups over a finite field

G

Normal coverings of linear groups

be a finite group, and let

ON THE DISTRIBUTION OF CONJUGACY CLASSES BETWEEN THE COSETS OF A FINITE GROUP IN A CYCLIC EXTENSION

\kappa(G)

PERFECT COMMUTING GRAPHS

be the probability that elements

Nilpotent covers and non-nilpotent subsets of finite groups of Lie type

g

A formal identity involving commuting triples of permutations

,

Cycle index methods for finite groups of orthogonal type in odd characteristic

h\in G