We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered p-dimensional ground-state sector. The index is fractional, with the denominator given by p. In particular, this yields a short proof of the quantization of the Hall conductance and of the Lieb-Schulz-Mattis theorem. In the case that the index is not an integer, the argument provides an explicit construction of Wilson loop operators exhibiting a nontrivial braiding that can be used to create fractionally charged Abelian anyons.