AbstractLet$G(q)$be a finite Chevalley group, where$q$is a power of a good prime$p$, and let$U(q)$be a Sylow$p$-subgroup of$G(q)$. Then a generalized version of a conjecture of Higman asserts that the number$k(U(q))$of conjugacy classes in$U(q)$is given by a polynomial in$q$with integer coefficients. In [S. M. Goodwin and G. Röhrle,J. Algebra321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of$k(U(q))$. By implementing it into a computer program using$\mathsf{GAP}$, they were able to calculate$k(U(q))$for$G$of rank at most five, thereby proving that for these cases$k(U(q))$is given by a polynomial in$q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of$k(U(q))$for finite Chevalley groups of rank six and seven, except$E_7$. We observe that$k(U(q))$is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write$k(U(q))$as a polynomial in$q-1$, then the coefficients are non-negative.Under the assumption that$k(U(q))$is a polynomial in$q-1$, we also give an explicit formula for the coefficients of$k(U(q))$of degrees zero, one and two.
Conjugacy Classes in Sylowp-Subgroups of Finite Chevalley Groups in Bad Characteristic
