Property Testing of Abelian Group Operations

Given an n n table of a cancellative operation on a domain of size n, we investigate the complexity of determining whether is close (equal on most pairs of inputs) to an associative, commutative, and cancellative group operation 0. We show that one can perform such a test in O(n) time. In contrast, quadratic time is necessary and sufficient to test that a given operation is cancellative, associative, and commutative. We give a sub-quadratic algorithm for the case when is not known to be cancellative. Our techniques for the case when is not known to be cancellative were later used by [EKK+97] to test that a function is associative in the same case. Furthermore, we show how to compute 0 in constant time, given access to . We show that our simple test can be used to quickly check the validity of tables of abelian groups and fields. Another application of our results is to testing programs that compute functions which are solutions to certain functional equations.