Let $G=[n]$ and let an unknown binary operation $*:G\times G\to G$ be given by oracle access to its multiplication table. The tester is given $n$ explicitly as part of its input. A query chooses $x,y\in G$ and receives $x*y$. We study randomized adaptive two-sided property testers for deciding whether $(G,*)$ is a group. An input is considered far from the property if one must change at least an $\epsilon$-fraction of the $n^2$ table entries in order to obtain any group operation on the same labeled set.

We prove three results. First, there is a tester using $$O\left(n+\frac1\epsilon\right)$$
queries and $\tilde O\left(n+1/\epsilon\right)$ time. In fact, this tester has one-sided error. This improve the Goldreich-Tauber tester~\cite{GoldreichTauber25}, that uses $\tilde O(n/\epsilon)$ queries.

Second, if computational efficiency is ignored, a dictionary-and-locator construction combined with an action-consistency test on an expanding Cayley graph gives a tester using
$$\tilde O\left(\sqrt{\frac n\epsilon}+\frac1\epsilon\right)$$
queries. This is the first query complexity bound for this problem that is sublinear in $n$.

Third, there are infinitely many values of $n$ for which every adaptive randomized two-sided tester needs $$\Omega \left(n^{1/5}+\frac1\epsilon\right)$$
queries. This improves on the $\Omega(\log n)$ lower bound of Goldreich and Tauber. The lower bound uses two randomly relabeled abelian groups of almost equal orders, embeds them into multiplication tables on the same $n$-element set, and couples arbitrary adaptive query transcripts.