Beating the Generator-Enumeration Bound for pp-Group Isomorphism

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G cong H. For several decades, the n^(log_p 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 the first 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) log_p 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 n^(O(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) log_p 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.Comment: 15 pages. This is an updated and improved version of the results for p-groups in arXiv:1205.0642 and TR11-052 in ECC

Algorithms for group isomorphism via group extensions and cohomology

The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N is abelian, this strategy is well-known. Our first contribution is to extend this strategy to handle the case when N is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. (ICALP 2012), namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on a $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation, with some new result