Computing modular and projective character degrees of soluble groups

Abstract

Let G be a finite soluble group and r a rational prime or zero. Let Z be a central cyclicsubgroup of G , if r >0, then the order of Z is relatively prime to r . Let F be an algebraically closed field of characteristic r . Let λ be a faithful linear character of Z in F . Such a λ gives rise to a factor system f for H = G/Z and any factor system for H in F so arises. An algorithm for determining the degrees of those irreducible representations of G , which restrict to Z to give the scalar representation, λ , is presented. If Z is the trivial subgroup, the algorithm can be used to compute the degrees of all FG -irreducibles (together with multiplicities); in particular, the number of conjugacy classes of G and for any prime r (>0), the number of r -regular conjugacy classes of G are determined. If Z is nontrivial, the same results are obtained for the twisted group algebra F f H with respect to f . The starting point is a power-commutator presentation for G ; only the supposed characteristic r of field F is used and the calculations are performed in sections of G . Clifford's theorem is used as the basic reduction tool.