What is the fastest algorithm to check whether input table is a group?

Question

Given an input table with binary operation $\circ$. I want to check whether given input table represents a group or not.

I need to verify the four axioms of a group.

1. The identity I can find it by just scanning the first row of the given table. -----$\mathcal{O}(|G|)$ running time.
2. Find the closure operation ie.e for any $x,y \in G$, I need to show that $x \circ y \in G $.------$\mathcal{O}(|G|^2)$
3. An inverse of an element.----$\mathcal{O}(|G|^2)$ Running time.
4. Associativity operation. -----$\mathcal{O}(|G|^3)$

Which gives me a $\mathcal{O}(|G|^3)$ running time algorithm to check whether the input table corresponds to some group.

Question : What is the fastest algorithm to check whether the input table is a group?

Model of computation is RAM, which computing $x \circ y$ takes constant time.

Answer 1

There is a non-trivial randomized algorithm that can solve this in $O(n^2 \log (1/\delta))$ time, where $\delta>0$ is the desired error probability. See

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