Where isomorphisms of group algebras fail to lift

TL;DR

This paper investigates the failure of lifting isomorphisms of group algebras across different rings, providing new counterexamples that deepen understanding of the Modular Isomorphism Problem.

Contribution

It demonstrates that certain non-isomorphic groups have non-isomorphic group algebras over the ring , extending previous counterexamples to a broader class of rings.

Findings

Counterexamples over  are identified.

Non-isomorphic groups have isomorphic algebras over .

The failure of lifting isomorphisms occurs already at .

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 $\mathbb{Z}/4\mathbb{Z}$.