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

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.