  Dirac fermions coupled to gauge fields can exhibit the chiral anomaly even on
a finite spatial lattice. A careful description of this phenomenon yields new
insights into the nature of spin-charge relations and on-site symmetries
(symmetries that are gauged by placing gauge fields on all links of the
lattice). One notable result is that only sufficiently small symmetry groups
can act on-site in a system with finitely many degrees of freedom. Symmetries
that break this rule either cannot be gauged on an arbitrary lattice, or the
gauging decouples the matter fields from gauge fields. These "anomalies" are
not quantum in nature, and they are diagnosed geometrically, by the volume of
the spatial manifold. The familiar particle number ${\mathrm U}(1)$ exhibits
this kind of anomalous behavior in any finite fermion theory. The chiral
anomaly in such a finite system instead manifests itself most simply after
gauging the ${\mathbb Z}_2$ fermion parity.
