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We give new polynomial-time algorithms for testing isomorphism of a class ofgroups given by multiplication tables (GpI). Two results (Cannon & Holt, J.Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reducesto the following: given groups G, H with characteristic subgroups of the sametype and isomorphic to $\mathbb{Z}_p^d$, and given the coset of isomorphisms$Iso(G/\mathbb{Z}_p^d, H/\mathbb{Z}_p^d)$, compute Iso(G, H) in time poly(|G|).Babai & Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of$G/\mathbb{Z}_p^d$ is trivial. In this paper, we solve the preceding problem inthe so-called "tame" case, i.e., when a Sylow p-subgroup of $G/\mathbb{Z}_p^d$is cyclic, dihedral, semi-dihedral, or generalized quaternion. These casescorrespond exactly to the group algebra$\overline{\mathbb{F}}_p[G/\mathbb{Z}_p^d]$ being of tame type, as in thecelebrated tame-wild dichotomy in representation theory. We then solve newcases of GpI in polynomial time.  Our result relies crucially on the divide-and-conquer strategy proposedearlier by the authors (CCC 2014), which splits GpI into two problems, one ongroup actions (representations), and one on group cohomology. Based on thisstrategy, we combine permutation group and representation algorithms with newmathematical results, including bounds on the number of indecomposablerepresentations of groups in the tame case, and on the size of their cohomologygroups.  Finally, we note that when a group extension is not tame, the precedingbounds do not hold. This suggests a precise sense in which the tame-wilddichotomy from representation theory may also be a dividing line between the(currently) easy and hard instances of GpI.

Polynomial-time isomorphism test of groups that are tame extensions