Clifford's theorem for bricks

Abstract

Let $G$ be a finite group, $N$ a normal subgroup of $G$, and $k$ a field of characteristic $p>0$. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup $N$ in $G$ is a $p$-power.