Verifying Groups in Linear Time

Consider the following problem: Given an n × n multiplication table, decide whether it is a Cayley multiplication table of a group. Among deterministic algorithms for this problem, the best known algorithm is implied by F. W. Light's associativity test (1949) and has running time of O(n²log n). Allowing randomization. the best known algorithm has running time of O(n²log(1/Δ)), where Δ > 0 is the error probability of the algorithm (Rajagopalan and Schulman, FOCS 1996, SICOMP 2000). In this work, we improve upon both of the above known algorithms. Specifically, we present a deterministic algorithm for the above problem whose running time is O(n²). This performance is optimal up to constants. A central tool we develop is an efficient algorithm for finding a subset A of a group G satisfying A²=G while |A|=O(√{|G|}).