Where isomorphisms of group algebras fail to lift

Abstract

Counterexamples to the Modular Isomorphism Problem were discovered recently. These are non-isomorphic finite $2$-groups $G$ and $H$ that have isomorphic group algebras over the field $\mathbb{Z}/2\mathbb{Z}$ and non-isomorphic group algebras over the $2$-adic integers $\mathbb{Z}_2$. We show that the groups $G$ and $H$ already have non-isomorphic group algebras over the ring $\mathbb{Z}/4\mathbb{Z}$.