  It is well known that if G is a finite group then the group of endotrivial
modules is finitely generated. In this paper we prove that for an arbitrary
finite group scheme G, and for any fixed integer n > 0, there are only finitely
many isomorphism classes of endotrivial modules of dimension n. This provides
evidence to support the speculation that the group of endotrivial modules for a
finite group scheme is always finitely generated. The result also has some
applications to questions about lifting and twisting the structure of
endotrivial modules in the case that G is an infinitesimal group scheme
associated to an algebraic group.
