Recognising nilpotent groups

In this paper we prove some results concerning how much information about the structure of a finite group can be gained from knowledge of the set of sizes of the conjugacy classes. We give examples to show that in general nilpotency cannot be recognised. However we show that for metabelian A-groups it is possible to identify nilpotent groups. We also prove that if an A-group G has a conjugacy class of size a2, where a2 is the maximal power of 2 to divide the size of any conjugacy class, then the group is soluble.