Beating the generator-enumeration bound for p-group isomorphism

Abstract

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G≅H. For several decades, the nlogp⁡n+O(1) generator-enumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show an improvement over the generator-enumeration bound for p-groups, which are believed to be the hard case of the group isomorphism problem. We start by giving a Turing reduction from group isomorphism to n(1/2)logp⁡n+O(1) instances of p-group composition-series isomorphism. By showing a Karp reduction from p-group composition-series isomorphism to testing isomorphism of graphs of degree at most p+O(1) and applying algorithms for testing isomorphism of graphs of bounded degree, we obtain an nO(p) time algorithm for p-group composition-series isomorphism. Combining these two results yields an algorithm for p-group isomorphism that takes at most n(1/2)logp⁡n+O(p) time. This algorithm is faster than generator-enumeration when p is small and slower when p is large. Choosing the faster algorithm based on p and n yields an upper bound of n(1/2+o(1))log⁡n for p-group isomorphism.