In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank 2 elementary abelian ℓ-subgroups in any finite group of Lie type, for any prime ℓ, which may be of independent interest.

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Abstract. We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved for the case of p-nilpotent groups.

Abstract. We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that for n ≥ p 2 the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups.

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n p 2 , the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n p 2 + p and has rank 2 if p 2 n < p 2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano. = Hom B (µ, ResHowever, a direct computation shows that Res J A J A ∩B χ is not trivial. So we have a contradiction to the assumption that Y A is endotrivial.