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For      M M       any liftable module of a group algebra of a finite group      G G       over a field of characteristic      p p       and      x ∈  G x \in G       any      p p      -element, we discuss connections between the restriction of      M M       to      ⟨ x ⟩ \left \langle x \right \rangle       and that of      χ  \chi       to      ⟨ x ⟩ \left \langle x \right \rangle      , where      χ  \chi       is the character of      M M      . In particular we get a lower bound for the number of Jordan components of this restriction of      M M       in terms of      χ  \chi       restriction. For      M 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.

On centralizers of 𝑝-elements in indecomposable modules