$G$-algebras, group graded algebras, and Clifford extensions of blocks

TL;DR

This paper generalizes Dade's results by associating a group extension to a block of a K-interior H-algebra and proving its isomorphism to an extension derived from the Brauer homomorphism, offering an alternative perspective.

Contribution

It introduces a new approach to associating group extensions with blocks of K-interior H-algebras and proves their isomorphism to extensions from the Brauer homomorphism, extending Dade's work.

Findings

Established an isomorphism between two group extensions associated with blocks.

Provided a new framework for understanding block extensions in group algebras.

Generalized Dade's results on block extensions.

Abstract

Let $K$ be a normal subgroup of the finite group $H$. To a block of a $K$-interior $H$-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to a block given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results "Block extensions" Section 12.