Group isomorphism is nearly-linear time for most orders

We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. This improves significantly over the general quasi-polynomial time complexity and shows that group isomorphism can be tested efficiently for almost all group orders. We also show that in nearly-linear time it can be decided whether a multiplication table describes a group; this improves over the known super-linear complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model, but we give the implications to a RAM model in the promise hierarchy as well.