On Higman`s Conjecture

Abstract

Let Gn be the subgroup of GLn(q) consisting of the upper unitriangular matrices of size nxn over Fq. In 1960, G. Higman conjectured that the number of conjugacy classes of Gn, denoted by r(Gn), was given by a polynomial in q with integer coefficients. This has been verified for nn, r(Gn) can be expressed in terms of r(Gi), with i<n, and r(Tn), where Tn is the subset of primitive canonical matrices of Gn. Moreover, we determinate the expression of r(Tn) modulo (q-1)[(+1)/2]+3 and we prove that $r(Tn) mod(q-1)[(+1)/2]+3 is a polynomial in q with integer coefficients.