Testing Group Commutativity in Constant Time

Lipton and Zalcstein presented a constant time algorithm for testing if a group is abelian in 12. However, the reference only contains a short abstract without proof. In this paper, we give a self contained proof for an n 2 3 lower bound for the number of pairs (a, b) of elements with ab 6= ba in every non-commutative group of size n. It implies a constant time randomized algorithm that tests if a group of n elements is commutative. Our lower bound for the number of non-commutative pairs (a, b) (ab 6= ba) in a non-commutative group of size n has a generalized format (p−1)(q−1)n pq , where p > 1 is the least integer divisor of n, and q > p is the second least integer divisor of n. keywords: abelian group, randomization, algorithm.