On endo-trivial modules for p-solvable groups

Abstract

In this note, we will prove a conjecture of Carlson et al. [1], thus resolving the question left open in [1] and completing the classification of torsion endo-trivial modules for p-solvable groups. Namely, we will prove that if G is a finite p-nilpotent group which contains a non-cyclic elementary Abelian p-subgroup and k is an algebraically closed field of characteristic p, then all simple endo-trivial kG-modules are 1-dimensional. In fact, we do rather more: we prove the analogous result directly in the case that G is p-solvable and contains an elementary Abelian p-subgroup of order p2. Carlson, Mazza and Thevenaz had reduced the proof of this result for p-solvable G to the p-nilpotent case (and had proved the result in the solvable case), but our method is somewhat different. Our proof does require the classification of finite simple groups. Specifically, we require the well-known fact that the outer automorphism group of a finite simple group of order prime to p has cyclic Sylow p-subgroups (see, for example, Theorem 7.1.2 of Gorenstein et al. [4]). Let us recall that a kG-module M is endo-trivial if M ⊗ M∗ ∼= k ⊕ N , where N is a projective kG-module. If |G| is divisible by p, then any endo-trivial kG-module has dimension prime to p.The vertex of any indecomposable endo-trivial kG-module is a Sylow p-subgroup of G. We remark that if M is an endo-trivial kG-module which is not 1-dimensional, then a Sylow p-subgroup of G acts faithfully on M ⊗ M∗ and hence acts faithfully on M.