Code Equivalence and Group Isomorphism

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 worstcase guarantees on a n o(log n) -time algorithm that tackles both aspects of GpI-actions and cohomology-simultaneously.Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were n O(log n) , even for groups with a central radical of constant size, such as Rad(G) = Z(G) = Z 2 . To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work. * The introduction may serve as an extended abstract:

show abstract

“…In the context of GpI, it has also been used successfully in the polynomial-time algorithm for semisimple groups [BCGQ11,BCQ12].…”

Section: The Babai-beals Filtrationmentioning

confidence: 99%

Algorithms for Group Isomorphism via Group Extensions and Cohomology

Grochow

Qiao

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

Self Cite

View full text Add to dashboard Cite

show abstract

“…2 The fundamental property of the centroid is that ı is bilinear with respect to the ring Cen.ı/. Centroids in this form arose to describe direct product decompositions of p-groups, cf.…”

Section: Centroid and Derivation Refinementsmentioning

confidence: 99%

“…In these rings the Jacobson radical is trivial. Likewise, the centroid (which embeds in the adjoint 2 The centroid for L s =L ? ring) is Z.…”

Section: Concrete Examplesmentioning

confidence: 99%

See 1 more Smart Citation

More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

Wilson¹

2013

Journal of Group Theory

View full text Add to dashboard Cite

Abstract. We introduce three families of characteristic subgroups that refine the traditional verbal subgroup filters, such as the lower central series, to an arbitrary length. We prove that a positive logarithmic proportion of finite p-groups admit at least five such proper nontrivial characteristic subgroups whereas verbal and marginal methods explain only one. The placement of these subgroups in the lattice of subgroups is naturally recorded by a filter over an arbitrary commutative monoid M and induces an M -graded Lie ring. These Lie rings permit an efficient specialization of the nilpotent quotient algorithm to construct automorphisms and decide isomorphism of finite p-groups. As a demonstration, we identify some large families of p-groups that are worst-case examples for the traditional nilpotent quotient algorithm but run in polynomial time when using our new filters.

show abstract

“…For example, the natural isomorphism problem for permutation groups, which asks for a permutational isomorphism between two given groups, reduces to the normalizer problem in polynomial time [Luk91]. Similarly, the isomorphism problem for linear codes, also known as code equivalence, is polynomial time reducible to permutational isomorphism and thus to the normalizer problem [BCGQ11].…”

Section: Introductionmentioning

confidence: 99%