Endotrivial Modules in the Cyclic Case
نویسندگان
چکیده
The purpose of this note is to determine all endotrivial modules in prime characteristic p, for a finite group having a cyclic Sylow p-subgroup. In other words, we describe completely the group of endotrivial modules in that case. The endotrivial modules are a family of finitely generated modules for a group algebra of a finite group which appear naturally in modular representation theory. Several contributions towards a general classification have already been obtained (cf. [2], [3], [4], [5], [6], [7] and [8]). We write T (G) for the group of endotrival modules for the finite group G. Assume that G has a cyclic Sylow p-subgroup P and let H be the normalizer in G of the unique subgroup of P of order p. We prove here that induction and restriction induce inverse isomorphisms T (G) ∼= T (H) and that T (H) is generated by one-dimensional representations of H together with the class of the first syzygy of the trivial module, subject to a single explicit relation.
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