The Modular Isomorphism Problem for small groups – revisiting Eick's algorithm

Abstract

We study the Modular Isomorphism Problem for groups of small order based on an improvement of an algorithm described by Eick. Our improvement allows to determine quotients $I\left(kG\right)/I{\left(kG\right)}^m$ of the augmentation ideal without first computing the full augmentation ideal $I\left(kG\right)$. Our computations yield a positive answer to the MIP for groups of order 3⁷ and strongly reduce the cases that need to be checked for groups of order 5⁶. We also show that the counterexamples to the Modular Isomorphism Problem found recently by García-Lucas, Margolis and del Río are the only 2- or 3-generated counterexamples of order 2⁹. Furthermore, we provide a proof for an observation of Bagiński, which is helpful in eliminating computationally difficult cases.