What is the fastest algorithm to check whether input table is a group?
There is a non-trivial randomized algorithm that can solve this in $O(n^2 \log (1/\delta))$ time, where $\delta&gt;0$ is the desired error probability.  See

Verification of Identities.  Sridhar Rajagopalan and Leonard J. Schulman.  SIAM Journal on Computing, 29(4), pp.1155-1163.