Polynomial-Time Isomorphism Test of Groups that are Tame Extensions

Grochow, Joshua A.; Qiao, Youming

doi:10.1007/978-3-662-48971-0_49

“…In a follow-up work [GQ15], the current authors used the viewpoint of this paper to generalize the preceding from coprime extensions to so-called "tame" extensions. These are extensions of Z k p by Q (assuming Aut(Q) is known, e. g., by recursive divide-and-conquer) where the Sylow p-subgroups of Q are cyclic, or p = 2 and the Sylow 2-subgroups are dihedral, semi-dihedral, or generalized quaternion.…”

Section: Some Recent Results From the Point Of View Of The Main Lemmamentioning

confidence: 99%

“…In fact, the story behind [GQ15] is a perfect example of the utility of explicitly splitting GpI into Action Compatibility and Cohomology Class Isomorphism. Namely, independently, one of the current authors had solved Action Compatibility for the tame case, and the other had solved Cohomology Class Isomorphism for the tame case under the assumption that Action Compatibility could be solved; when they met in Chicago each was eager to tell the other of their result, asking if the other "half" of the problem could be solved.…”

Section: Some Recent Results From the Point Of View Of The Main Lemmamentioning

confidence: 99%

“…2.2]), but even the extensive body of work on practical algorithms led by Eick, Holt, Leedham-Green and O'Brien (e. g., [BEO02, ELGO02, BE99, CH03])-resulting in most of the functional algorithms in use today-was recently found [Wil14] to only improve the constant in the exponent, still resulting in a n Θ(log n) -time algorithm for the general case. The past few years have witnessed a resurgence of activity on worst-case guaranteed algorithms for this problem [LG09, BCGQ11, QST11, Wag11, LW12, BQ12, BCQ12, Ros13b, Ros13a, BMW15,GQ15].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Algorithms for Group Isomorphism via Group Extensions and Cohomology

Grochow

¹

,

Qiao

²

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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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

“…For groups of order n, the easy n log n+O(1) -time algorithm [FN70,Mil78] 1 for the general case of GpI has barely seen any asymptotic improvement over the past four decades; it was improved recently to n 1/4 log n+O(1) by Rosenbaum [Ros13a] (see [GR16,Sec. 2.2]), but even the extensive body of work on practical algorithms led by Eick, Holt, Leedham-Green and O'Brien (e. g., [BEO02, ELGO02, BE99, CH03])-resulting in most of the functional algorithms in use today-was recently found [Wil14] to only improve the constant in the exponent, still resulting in a n Θ(log n) -time algorithm for the general case. The past few years have witnessed a resurgence of activity on worst-case guaranteed algorithms for this problem [LG09, BCGQ11, QST11, Wag11, LW12, BQ12, BCQ12, Ros13b, Ros13a, BMW15,GQ15].…”

Section: Introductionmentioning

confidence: 99%

“…This suggests the presence of abelian normal subgroups as a bottleneck. With this in mind, Babai and Qiao [BQ12] developed a polynomial-time algorithm for a special class of non-nilpotent solvable groups, building on [LG09,QST11]; this was recently extended by the present authors to the so-called groups of tame extensions [GQ15]. In [LW12], Lewis and Wilson made intriguing progress on p-groups: They gave a polynomial-time algorithm for quotients of generalized Heisenberg groups, a decently large subclass of p-groups of class 2.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Group Isomorphism via Group Extensions and Cohomology

Grochow¹,

Qiao²

2017

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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