ENDO-TRIVIAL MODULES FOR FINITE GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUP

SHIGEO KOSHITANI AND CAROLINE LASSUEUR

Abstract. Let k be an algebraically closed field of characteristic p > 0 and G a finite
group. We provide a description of the torsion subgroup T T (G) of the finitely generated
abelian group T (G) of endo-trivial kG-modules when p = 2 and G has a dihedral Sylow
2-subgroup P. We prove that, in the case |P| ≥ 8, TT(G) ∼= X(G) the group of
one-dimensional kG-modules, except possibly when G/O2′(G) ∼= A6, the alternating
group of degree 6; in which case G may have 9-dimensional simple torsion endo-trivial
modules. We also prove a similar result in the case |P | = 4, although the situation is more
involved. Our results complement the tame-representation type investigation of endo-
trivial modules started by Carlson-Mazza-Thevenaz [10] in the cases of semi-dihedral and
generalized quaternion Sylow 2-subgroups. Furthermore we provide a general reduction
result, valid at any prime p, to recover the structure of TT(G) from the structure of
TT(G/H), where H is a normal p′-subgroup of G.