  If chi is an irreducible character of a finite group G then the support of
chi is the subset of G on which chi does not vanish. In this note, we study the
supports of characters of certain classes of p-groups (a p-group is a finite
group of prime power order). We show that if chi is an irreducible character of
a metabelian p-group P, then the support of chi contains at most |P|/chi(1)^2
conjugacy classes, and we give a bound on the orders of the elements in the
support. We give more precise results for p-groups of nilpotence class at most
3 and groups of order dividing p^9.
