Group algebra extensions of depth one

Abstract

A ring extension $A\subseteq B$ is said to have depth one if $B$ is isomorphic to a direct summand of $A^n$ as an $\left(A,A\right)$-bimodule, for some positive integer $n$. We prove group-theoretic characterizations of this property in the case $kH\subseteq kG$, where $H$ is a subgroup of a finite group $G$ and $k$ is a field. We determine when the source algebra of a block of $kG$ with defect group $P$ is a depth-one extension of $kP$.