  Topological order in two-dimensional systems is studied by combining the
braid group formalism with a gauge invariance analysis. We show that flux
insertions (or large gauge transformations) pertinent to the toroidal topology
induce automorphisms of the braid group, giving rise to a unified algebraic
structure that characterizes the ground-state subspace and fractionally
charged, anyonic quasiparticles. Minimal ground state degeneracy is derived
without assuming any relation between quasiparticle charge and statistics. We
also point out that noncommutativity between large gauge transformations is
essential for the topological order in the fractional quantum Hall effect.
