Torsion-free endotrivial modules

Abstract

Abstract. Let G be a finite group and let T (G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part T F (G) of the group T (G) and look for generators of T F (G). We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that T F (G) can be generated by modules belonging to the principal block and we prove the conjecture in some cases.

Endotrivial modules : a reduction to p′-central extensions

“…Part (a) of Conjecture 5.1 is in fact a consequence of any of the two conjectures made in [CMT11b]. First, Conjecture 10.1 in [CMT11b] asserts that, if a group homomorphism φ : G → G induces an isomorphism between the corresponding p-fusion systems, then φ should induce an isomorphism T F (G ) ∼ → T F (G).…”

“…Part (a) of Conjecture 5.1 is in fact a consequence of any of the two conjectures made in [CMT11b]. First, Conjecture 10.1 in [CMT11b] asserts that, if a group homomorphism φ : G → G induces an isomorphism between the corresponding p-fusion systems, then φ should induce an isomorphism T F (G ) ∼ → T F (G). In the special case where φ is the quotient map φ : G → Q = G/O p (G), it is well-known that the fusion systems are isomorphic, so we would obtain the isomorphism T F (Q) ∼ → T F (G) of Conjecture 5.1 above.…”

“…In the special case where φ is the quotient map φ : G → Q = G/O p (G), it is well-known that the fusion systems are isomorphic, so we would obtain the isomorphism T F (Q) ∼ → T F (G) of Conjecture 5.1 above. This special case is explicitly mentioned at the end of Section 10 in [CMT11b].…”

“…Conjecture 9.2 in [CMT11b] asserts that the group T F (G) should be generated by endotrivial modules lying in the principal block. Since O p (G) acts trivially on any module lying in the principal block of G, such a module is inflated from Q, so the inflation map Inf G Q : T F (Q) −→ T F (G) in Conjecture 5.1 above should be an isomorphism.…”

The torsion group of endotrivial modules

“…They are modules which have universal deformation rings [13]. The endotrivial modules have been classified in case G is a p-group ( [11,12]) and various results have appeared since for some specific families of groups [4,5,6,7,8,9,10,18,19,20,17]. Recently, another line of research has developed that is concerned with the classification of all endotrivial modules which are simple [21,15,16].…”