  Mednykh proved that for any finite group G and any orientable surface S,
there is a formula for #Hom(pi_1(S), G) in terms of the Euler characteristic of
S and the dimensions of the irreducible representations of G. A similar formula
in the nonorientable case was proved by Frobenius and Schur. Both of these
proofs use character theory and an explicit presentation for \pi_1. These
results have been reproven using quantum field theory. Here we present a
greatly simplified proof of these results which uses only elementary topology
and combinatorics. The main tool is an elementary invariant of surfaces
attached to a semisimple algebra called a lattice topological quantum field
theory.
