### Graph Isomorphism is not AC^0 reducible to Group Isomorphism

We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits, where $n$ is the number of group elements. This improves the existing upper bound from \cite{Wolf 94} for the problems. In the previous upper bound the circuits...