COMPUTING MODULAR AND PROJECTIVE CHARACTER DEGREES OF SOLUBLE GROUPS

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 FfH 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. © 1990, Academic Press Limited. All rights reserved.