  In this paper we use Euclidean gravity to derive a simple formula for the
density of black hole microstates which transform in each irreducible
representation of any finite gauge group. Since each representation appears
with nonzero density, this gives a new proof of the completeness hypothesis for
finite gauge fields. Inspired by the generality of the argument we further
propose that the formula applies at high energy in any quantum field theory
with a finite-group global symmetry, and give some evidence for this
conjecture.
