# On the blockwise modular isomorphism problem

## TL;DR

This paper explores how Morita equivalence of blocks in finite groups influences the structure of defect groups, establishing new results for low defect and specific group classes, enhancing understanding of modular representation theory.

## Contribution

It proves that Morita equivalence class of a block with defect at most 3 determines its defect groups up to isomorphism, and extends similar results to certain cases in characteristic 0.

## Key findings

- Morita equivalence class determines defect groups for defect ≤ 3
- Results for metacyclic defect groups in characteristic 0
- Group algebra determines if G has abelian Sylow p-subgroups

## Abstract

As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita equivalence class of a block B of defect at most 3 determines the defect groups of B up to isomorphism. In characteristic 0 we prove similar results for metacyclic defect groups and 2-blocks of defect 4. In the second part of the paper we investigate the situation for p-solvable groups G. Among other results we show that the group algebra of G itself determines if G has abelian Sylow p-subgroups.