Breaking the n^{log n} Barrier for Solvable-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. The n log n barrier for group isomorphism has withstood all attacks -even for the special cases of p-groups and solvable groups -ever since the n log n+O(1) generator-enumeration algorithm. Following a framework due to Wagner, we present the first significant improvement over n log n by reducing group isomorphism to composition-series isomorphism which is then reduced to low-degree graph isomorphism. We show that group isomorphism is n (1/2) log p n+O(1) Turing reducible to composition-series isomorphism where p is the smallest prime dividing the order of the group. Combining our reduction with an n O(p) algorithm for p-group composition-series isomorphism, we obtain an n (1/2) log n+O(1) algorithm for p-group isomorphism. We then generalize our techniques from p-groups using Sylow bases to derive an n (1/2) log n+O(log n/ log log n) algorithm for solvable-group isomorphism. Finally, we relate group isomorphism to the collision problem which allows us replace the 1/2 in the exponents with 1/4 using randomized algorithms and 1/6 using quantum algorithms. arXiv:1205.0642v6 [cs.DS] 11 Dec 2013 Theorem 1.2. Group isomorphism is n (1/2) log p n+O(1) time deterministic Turing reducible to compositionseries isomorphism where p is the smallest prime dividing the order of the group.A composition series S is a tower of subgroups G 0 = 1 · · · G m = G such that no additional subgroups can be inserted. The factor groups G i+1 /G i are called the composition factors of S. We say that two composition seriesIn the composition-series isomorphism problem, we are given composition series S and S where the subgroups are specified by their corresponding subsets and must decide whether S ∼ = S .Wagner [39] showed a Karp reduction from composition-series isomorphism to low-degree graph isomorphism [26,6,5]. Using this reduction, he gave an n O(p) algorithm for p-group composition series isomorphism using an n O(d) algorithm 1 for isomorphism of graphs of degree at most d. Wagner's reduction also applies to general groups, but produces graphs of large degree when there are large composition factors. Wagner reduced the degrees by using a modification of the generatorenumeration algorithm to guess the action of the isomorphism on large composition factors; the graphs are then modified to enforce this mapping. This would have yielded an n O(log n/ log log n) algorithm for isomorphism of composition series for arbitrary groups. Unfortunately, the resulting graphs depend on which coset representatives are chosen so the algorithm is flawed. Wagner attempted to fix this error in two subsequent revisions [40,41]; unfortunately, these contain a more subtle manifestation of the same flaw. Wagner retracted his attempted fix in the latest revision of his paper [42] following further discussions with the author (see Appendix E.) However, we will show that Wagner's generator-fixing trick can still be salvaged in a s...

Analysis of an Exhaustive Search Algorithm in Random Graphs and the $n^{c\log n}$-Asymptotics

“…-Algorithmics: isomorphism testing (see Babai and Qiao (2012); Grošek and Sýs (2010); Huber (2011);Miller (1978); Rosenbaum (2012)), autocorrelations of strings (see Guibas and Odlyzko (1981); Rivals and Rahmann (2003)), information theory (see Abu-Mostafa (1986)), random digital search trees (see Drmota (2009)), population recovery (see Wigderson and Yehudayoff (2012)), and asymptotics of recurrences (see Knuth (1966); O' Shea (2004));…”

Untitled

“…Indeed, Group Isomorphism is not currently known to be testable in time n o(log n) (cf. [4,6,5,7]). Therefore, if our bound on k is not very far from being tight, say the result stated in Remark 1.2 would fail if we replace 2 log 3 n by O((log n) 0.99 ), this would mean progress on the complexity of the Group Isomorphism problem.…”

A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension

“…In fact the only improvement for the worst case complexity over Tarjan's classic n log(n)+O (1) algorithm are n 1 c •log(n)+O (1) algorithms with a small constant c depending on the model of computation (randomization, quantum computing etc.) [23,26,27]. There is however a nearly-linear time algorithm that solves group isomorphism for most orders [10].…”