# On Picard groups of blocks with normal defect groups

## TL;DR

This paper proves that all Morita auto-equivalences of blocks with normal abelian defect groups and abelian inertial quotient have linear sources, advancing understanding of their structure and symmetries.

## Contribution

It establishes that every Morita auto-equivalence of such blocks possesses linear sources, improving previous results by Zhou, Boltje, Kessar, and Linckelmann.

## Key findings

- All Morita auto-equivalences have linear sources.
- The result applies to blocks with normal abelian defect groups.
- It generalizes and improves earlier findings.

## Abstract

Let $b$ be a block with normal abelian defect group and abelian inertial quotient. We prove that every Morita auto-equivalence of $b$ has linear source. We note that this improves upon results of Zhou and also Boltje, Kessar and Linckelmann.