Group isomorphism is nearly-linear time for most orders

We show that there is a dense set $\Upsilon\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \Upsilon$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic groups of order $n$. This improves significantly over the general $n^{O(\log n)}$-time complexity and shows that group isomorphism can be tested efficiently for almost all group orders $n$. We also show that in time $O(n^2 (\log n)^d)$ it can be decided whether an $n\times n$ multiplication table describes a group; this improves over the known $O(n^3)$ complexity.