  Let G be a connected, reductive group over an algebraically closed field of
good characteristic. For u in G unipotent, we describe the conjugacy classes in
the component group A(u) of the centralizer of u. Our results extend work of
the second author done for simple, adjoint G over the complex numbers.
  When G is simple and adjoint, the previous work of the second author makes
our description combinatorial and explicit; moreover, it turns out that
knowledge of the conjugacy classes suffices to determine the group structure of
A(u). Thus we obtain the result, previously known through case-checking, that
the structure of the component group A(u) is independent of good
characteristic.
