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Simple endotrivial modules for linear, unitary and exceptional groups. (English) Zbl 1327.20008
The paper under review is a continuation of a paper by C. Lassueur, G. Malle and E. Schulte [J. Reine Angew. Math. (to appear)]. Motivated by a result of G. R. Robinson [Bull. Lond. Math. Soc. 43, No. 4, 712-716 (2011; Zbl 1234.20003)], the aim is to classify the simple endotrivial modules of the quasi-simple finite groups. Here the authors focus their attention on special linear and unitary groups, where they obtain a complete classification, and on exceptional groups of Lie type.
In their earlier paper with Schulte, the authors conjectured that a quasi-simple finite group with a faithful simple endotrivial module in characteristic $$p$$ has $$p$$-rank at most $$2$$. In the present paper, the authors verify this conjecture. As another consequence of their results, the authors show that the principal $$p$$-block of a finite simple group with noncyclic Sylow $$p$$-subgroup cannot have Loewy length $$4$$, for $$p>2$$. This result is motivated by a paper by S. Koshitani, B. Külshammer and B. Sambale [Math. Proc. Camb. Philos. Soc. 156, No. 3, 555-570 (2014; Zbl 1329.20009)].

##### MSC:
 20C20 Modular representations and characters 20C33 Representations of finite groups of Lie type 20C34 Representations of sporadic groups 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
CHEVIE; GAP
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