Endotrivial Modules over Groups with Quaternion or Semi-dihedral Sylow 2-subgroup

Suppose that G is a finite group and that k is a field of characteristic p. Endotrivial kG-modules appear in a natural way in many areas surrounding local analysis of finite groups. They were introduced by Dade [14] who classified them in the case that G is an abelian p-group. A complete classification of endotrivial modules over the modular group rings of p-groups was completed just a few years ago [5, 10, 11, 12]. The class of all endotrivial modules for a given group G gives rise to an abelian group T (G) (with respect to the tensor product). This group is finitely generated and carries with it all of the information of the classification. The group T (G) is of interest because it is an important part of the Picard group of self-equivalences of the stable category of finitely generated kG-modules. The so-called self-equivalences of Morita type are induced by tensoring with endotrivial modules. For this reason, it is of interest to extend the classification beyond p-groups to general finite groups. Some progress has been made in that direction [6, 7, 8, 9, 21]. In this paper we consider two out-lying situations where the answer to a different sort of problem is sought. In the classification of endotrivial modules over p-groups, there are exactly two cases in which the group T (P ) of endotrivial modules for a non-cyclic p-group P has torsion elements. The two cases case occur when p = 2 and P is either quaternion (meaning ordinary or generalized quaternion) or semi-dihedral. For a group G having such a P as its Sylow 2-subgroup, the question is whether the restriction map T (G) → T (P ) is surjective. Specifically, we need to know if the torsion elements in T (P ) are in the image of the restriction. Do these modules lift or extend in some way from P to G? In this paper we show that the answer is yes, the restriction map is surjective. In the course of the investigation we are able to find much more information about the structure of T (G) and about the modules themselves. The only other case in which T (P ) has torsion elements occurs when P is cyclic, and this case was treated in [21]. It is somewhat surprising that the two cases require very different methods. In the situation where the Sylow 2-subgroup P of G is quaternion and the unique involution in P is central in G, we use a general method for finding exotic endotrivial modules as subquotients of Ω(k), the second syzygy of the the trivial module k. This method has been used in earlier papers [5, 7]. There are two means for extending this result to general groups with quaternion Sylow 2-subgroups. One involves invoking the Brauer-Suzuki Theorem [4] on the stucture of such group. The more elementary method is to note that the centralizer of the involution of P is a strongly 2-embedded subgroup of G and we can apply a theorem of [21]. These results appear in Sections 3 and 4, after