Fitting heights of odd-order groups with few character degrees

Abstract

If G is any finite solvable group having a normal Sylow 2-subgroup (in particular, if | G| is odd) and satisfying |cd( G)|⩾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.