Groups whose irreducible character degrees are ordered by divisibility

This paper concerns the class of finite groups whose complex irreducible character degrees can be linearly ordered by divisibility. It is known that such a group has a Sylow tower. By analyzing the structure of a group in the class whose order is divisible by just two primes, we are able to obtain information on the Sylow subgroups of any group in the class. We classify the two-prime groups in the class Except for certain exceptional pairs of primes, one of which is always 2. Such a group G of order p a q* either has a normal abelian Sylow ^-subgroup or H = G/O P (G) has a non-abelian Sylow ^-subgroup Q and each p-element of H induces a fixedpoint-free automorphism of Q !.