{
  "id": "1310.3647",
  "version": "v2",
  "published": "2013-10-14T12:23:35.000Z",
  "updated": "2014-10-09T08:22:58.000Z",
  "title": "Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups",
  "authors": [
    "Caroline Lassueur"
  ],
  "comment": "17 pages. Changes from (v1): S. Koshitani was added as an author. This is an improvement of (v1) focusing on the group of endotrivial modules for finite groups with Klein-four Sylow 2-subgroups. The last section was removed",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the group structure of $T(G)$ is known for the cases where a Sylow $p$-subgroup $P$ of $G$ is cyclic, semi-dihedral and generalized quaternion. We investigate $T(G)$, and more accurately, its torsion subgroup $TT(G)$ for the case where $P$ is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which $TT(G) = f^{-1}(X(N_{G}(P)))$ holds, where $f^{-1}(X(N_{G}(P)))$ denotes the abelian group consisting of the $kG$-Green correspondents of the one-dimensional $kN_{G}(P)$-modules. We show that the lift to characteristic zero of any indecomposable module in $TT(G)$ affords an irreducible ordinary character. Furthermore, we show that the property of a module in $f^{-1}(X(N_{G}(P)))$ of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-14T12:23:35.000Z",
      "title": "A note on Auslander-Reiten quiver methods for endo-trivial modules",
      "abstract": "The aim of the present note is to use Auslander-Reiten quiver techniques, on the one hand to describe the structure of the group of endo-trivial modules for classes of groups with Klein-four Sylow subgroups, and on the other hand to discard the existence of simple endo-trivial modules for alternating groups, symmetric groups and groups of Lie type in their defining characteristic, providing us with alternative proofs of results obtained in arxiv:1305.3466 and links with previous work on endo-trivial modules by Carlson-Mazza-Nakano.",
      "comment": "9 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-09T08:22:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C20"
    ],
    "keywords": [
      "finite group",
      "klein-four sylow",
      "endo-trivial modules",
      "group algebra",
      "irreducible ordinary character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.3647K"
    }
  }
}