  Let \chi be an irreducible character of the finite group G. If g is an
element of G and \chi(g) is not zero, then we conjecture that the order of g
divides |G|/\chi(1). The conjecture is a generalization of the classical fact
that irreducible p-projective characters vanish on p-singular elements, since
the latter is equivalent to saying that if \chi(g) is not zero then the square
free part of the order of g divides |G|/\chi(1). We prove some partial results
on the conjecture; in particular, we show that the order of g divides
(|G|/\chi(1))^2. Using these results, we derive some bounds on heights of
characters. We also pose a related conjecture concerning congruences satisfied
by central character values.
