Polynomial-time isomorphism test of groups that are tame extensions (Extended abstract)

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Conference Proceeding
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2015, 9472 pp. 578 - 589
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© Springer-Verlag Berlin Heidelberg 2015. We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the following: given groups G,H with characteristic subgroups of the same type and isomorphic to ℤdp , and given the coset of isomorphisms Iso(G/ℤdp ,H/ℤdp), compute Iso(G,H) in time poly(|G|). Babai&Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of G/ℤdp is trivial. In this paper, we solve the preceding problem in the so-called “tame” case, i. e., when a Sylow p-subgroup of G/ℤdp is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases correspond exactly to the group algebra (Formula presented.) being of tame type, as in the celebrated tame-wild dichotomy in representation theory. We then solve new cases of GpI in polynomial time. Our result relies crucially on the divide-and-conquer strategy proposed earlier by the authors (CCC 2014), which splits GpI into two problems, one on group actions (representations), and one on group cohomology. Based on this strategy, we combine permutation group and representation algorithms with new mathematical results, including bounds on the number of indecomposable representations of groups in the tame case, and on the size of their cohomology groups. Finally, we note that when a group extension is not tame, the preceding bounds do not hold. This suggests a precise sense in which the tame-wild dichotomy from representation theory may also be a key barrier to cross to put GpI into P.
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