Fitting heights of odd-order groups with few character degrees

If G is any finite solvable group having a normal Sylow 2-subgroup (in particular, if \G\ is odd) and satisfying \cd(G)\ greater than or equal to 5, where cd(G) is the set of ordinary irreducible character degrees of G, we show that the Fitting height of G does not exceed \cd(G)\ - 2. In case \cd(G)\ = 5, this upper bound on the Fitting height of G is best-possible. (C) 2003 Elsevier Inc. All rights reserved.