Centres of blocks of finite groups with trivial intersection Sylow $p$-subgroups

TL;DR

This paper investigates the structure of blocks in finite groups with trivial intersection Sylow p-subgroups, showing that certain stable equivalences do not induce algebra isomorphisms between centers, extending previous results.

Contribution

It generalizes known results about Suzuki groups to a broader class of groups with trivial intersection Sylow p-subgroups, analyzing their block centers and Morita equivalences.

Findings

Stable equivalences of Morita type do not induce algebra isomorphisms between centers.

Analysis of Loewy structures reveals differences in block centers.

Generalization of previous results to more groups with trivial intersection Sylow p-subgroups.

Abstract

For finite groups $G$ with non-abelian, trivial intersection Sylow $p$-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of $G$ and the centre of the Brauer correspondent. This was already known for the Suzuki groups; the result will be generalised to cover more groups with trivial intersection Sylow $p$-subgroups.