There are two positive integers naturally associated
with an irreducible character chi of a finite group G.
One of these is the degree chi(1), and the other is the
"determinantal order" o(chi), which is the order of
the determinant det(chi) in the group of linear
characters of G. Note that both of these integers
divide |G|.

Now assume that G is solvable and that pi is some
set of prime numbers, and consider the irreducible
characters chi of G such that both chi(1) and o(chi)
are pi-numbers. Also, since we want to consider
characters that behave well with respect to normal
subgroups, we impose the additional condition that
for every subnormal subgroup S of G, every
irreducible constituent theta of the restriction of
chi to S satisfies the same properties: theta(1) and
o(theta) are pi-numbers. Characters chi that satisfy
these conditions are said to be pi-special, and they
were first studied by D. Gajendragadkar.

We will discuss some of the remarkable and unexpected
properties of pi-special characters, and we will show
how they can be used to prove theorems that seem
difficult or impossible without them. One such result
is that if chi is a primitive irreducible character of a
solvable group G, then chi(1)^2 divides |G|, and
another is that if theta is an irreducible character of
a Hall subgroup H of a solvable group G and
theta(x) = theta(y) whenever x,y in H are conjugate
in G, then theta is the restriction of a character of G.

If time permits, we will also discuss other interactions
between the characters of a solvable group and sets
of primes, but I will not go into detail here.