Beating the GeneratorEnumeration 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 cong H. For several decades, the n^(log_p n + O(1)) generatorenumeration bound (where p is the smallest prime dividing the order of the group) has been the best worstcase result for general groups. In this work, we show the first improvement over the generatorenumeration bound for pgroups, 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 pgroup compositionseries isomorphism. By showing a Karp reduction from pgroup compositionseries 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 pgroup compositionseries isomorphism. Combining these two results yields an algorithm for pgroup isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time. This algorithm is faster than generatorenumeration 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 pgroup isomorphism.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.1755
 Bibcode:
 2013arXiv1312.1755R
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics
 EPrint:
 15 pages. This is an updated and improved version of the results for pgroups in arXiv:1205.0642 and TR11052 in ECCC