The classification of endo-trivial 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.

Torsion-free endotrivial modules

“…This is an example of a group in case (d) of the Berkovitch-Janko list. Theorem 3.1 improves the result given as Theorem 7.1 in [12] or Theorem 7.2 in [8] as follows. Theorem 3.3.…”

“…It is proved in [12] (using also [13]), but the statement now incorporates the improvement obtained in Theorem 3.1.…”

“…(Actually for both classes, we have |C P (S i )/S i | = 2.) Assume now that m i = 2 for some i ≥ 2, that is, C P (S i ) = S i × A where A is cyclic of order ≥ 4 (see Lemma 2.2 in [12]). For every j ≥ 2, we must prove that C P (S j ) ∼ = S j × C 2 q for some q ≥ 2 (in other words C P (S j )/S j is not quaternion).…”

“…, where Ω −m P/Si (k) denotes the relative syzygy of the trivial module, relative to the subgroup S i (see Section 3 in [12]). This is defined as the m-th cokernel in a minimal relative injective resolution of the trivial module.…”

The Dade group of a fusion system

“…By [12,Lemma 2.2], in that case, P has a unique central cyclic subgroup of order p, which we will denote by Z. Furthermore, any connected component of A d2 ðPÞ contains either all elementary abelian subgroups of rank at least 3 or consists of a single elementary abelian subgroup E of rank 2, in which case E is called isolated in A d2 ðPÞ.…”