Introduction

The aim of this note is to present some problems and also partial results in some cases, mainly on characters of p-groups. (In the last section we deal with a problem that consists in obtaining information about characters of a Sylow p-subgroup of an arbitrary group from information about the characters of the whole group.) This survey is far from being exhaustive. The topics included are strongly influenced by the author's interests in the last few years. There seems to be an increasing interest in the character theory of p-groups and we hope that this expository paper will encourage more research in the area. In the sixties I. M. Isaacs and D. S. Passman [17, 18] wrote two important papers that initiated the study of the degrees of the irreducible complex characters of finite groups (henceforth referred to as character degrees). The study of the influence of the set of character degrees on the structure of a group was taken up again in the eighties, in large part due to B. Huppert and his school. In particular, this has led to several papers dealing with the character degrees of important families of p-groups since the nineties (see [6, 8, 12, 28, 30, 32, 33, 34, 35, 36, 37]). Here we are mostly concerned with character degrees, but instead of studying particular families of p-groups, we intend to obtain general structural properties of groups according to their character degrees. Other problems on characters of p-groups appear in [25].