Character degrees of p-groups and pro-p groups

Abstract

In the 1970s, Isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p-group G with respect to the number of different irreducible character degrees of G. So far, there are just a few partial results for this conjecture. In this note, we say that a pro-p group G has property (I) if there is a real number D=D(G) that just depends on G such that for any open normal subgroup N, dl(G/N)⩽log2|cd(G/N)|+D. We prove that any p-adic analytic pro-p group has property (I). We also study the first congruence subgroup G of a classical Chevalley group G with respect to the local ring Fp〚t〛. We show that if Lie(G)(Fp) has a non-degenerated Killing form, then G has property (I).