Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

2015/07/06 by Koshitani, Shigeo, Lassueur, Caroline
#20C05 #20C20 #20C25 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1507.01406

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 TT(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/O_2'(G)≅ \mathfrakA₆, 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-Thévenaz 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.

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