On centralizers of $p$-elements in indecomposable modules

Abstract:

For $M$ any liftable module of a group algebra of a finite group $G$ over a field of characteristic $p$ and $x \in G$ any $p$-element, we discuss connections between the restriction of $M$ to $\left \langle x \right \rangle$ and that of $\chi$ to $\left \langle x \right \rangle$, where $\chi$ is the character of $M$. In particular we get a lower bound for the number of Jordan components of this restriction of $M$ in terms of $\chi$ restriction. For $M$ a permutation module, this bound is trivially an equality, and we derive several results by L. Scott, which hitherto have been considered relatively deep, in a very elementary and straightforward manner.