  Motivated by quantum simulation, we consider lattice Hamiltonians for
Yang-Mills gauge theories with finite gauge group, for example a finite
subgroup of a compact Lie group. We show that the electric Hamiltonian admits
an interpretation as a certain natural, non-unique Laplacian operator on the
finite Abelian or non-Abelian group, and derive some consequences from this
fact. Independently of the chosen Hamiltonian, we provide a full explicit
description of the physical, gauge-invariant Hilbert space for pure gauge
theories and derive a simple formula to compute its dimension. We illustrate
the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on
a small periodic lattice.
