  A charged particle in a uniform magnetic field in a two-dimensional torus has
a discrete noncommutative translation symmetry instead of a continuous
commutative translation symmetry. We study topology and symmetry of a particle
in a magnetic field in a torus of arbitrary dimensions. The magnetic
translation group (MTG) is defined as a group of translations that leave the
gauge field invariant. We show that the MTG on an n-dimensional torus is
isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x
Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible
unitary representations of the MTG on a three-torus and apply the
representation theory to three examples. We shortly describe a representation
theory for a general n-torus. The MTG on an n-torus can be regarded as a
generalization of the so-called noncommutative torus.
