{
  "id": "1511.07819",
  "version": "v1",
  "published": "2015-11-24T17:41:43.000Z",
  "updated": "2015-11-24T17:41:43.000Z",
  "title": "Artin transfer patterns on descendant trees of finite p-groups",
  "authors": [
    "Daniel C. Mayer"
  ],
  "comment": "39 pages, 9 figures, dedicated to Professor M. F. Newman",
  "categories": [
    "math.GR"
  ],
  "abstract": "Based on a thorough theory of the Artin transfer homomorphism \\(T_{G,H}:\\,G\\to H/H^\\prime\\) from a group \\(G\\) to the abelianization \\(H/H^\\prime\\) of a subgroup \\(H\\le G\\) of finite index \\(n=(G:H)\\), and its connection with the permutation representation \\(G\\to S_n\\) and the monomial representation \\(G\\to H\\wr S_n\\) of \\(G\\), the Artin pattern \\(G\\mapsto(\\tau(G),\\varkappa(G))\\), which consists of families \\(\\tau(G)=(H/H^\\prime)_{H\\le G}\\), resp. \\(\\varkappa(G)=(\\ker(T_{G,H}))_{H\\le G}\\), of transfer targets, resp. transfer kernels, is defined for the vertices \\(G\\in\\mathcal{T}\\) of any descendant tree \\(\\mathcal{T}\\) of finite \\(p\\)-groups. It is endowed with partial order relations \\(\\tau(\\pi(G))\\le\\tau(G)\\) and \\(\\varkappa(\\pi(G))\\ge\\varkappa(G)\\), which are compatible with the parent-descendant relation \\(\\pi(G)<G\\) of the edges \\(G\\to\\pi(G)\\) of the tree \\(\\mathcal{T}\\). The partial order enables termination criteria for the \\(p\\)-group generation algorithm which can be used for searching and identifying a finite \\(p\\)-group \\(G\\), whose Artin pattern \\((\\tau(G),\\varkappa(G))\\) is known completely or at least partially, by constructing the descendant tree with the abelianization \\(G/G^\\prime\\) of \\(G\\) as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns \\((\\tau(G),\\varkappa(G))\\) and explaining the stabilization, resp. polarization, of their components in descendant trees \\(\\mathcal{T}\\) of finite \\(p\\)-groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-24T17:41:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D15",
      "20F12",
      "20F14",
      "11R37"
    ],
    "keywords": [
      "descendant tree",
      "artin transfer patterns",
      "finite p-groups",
      "partial order enables termination criteria",
      "artin pattern"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151107819M"
    }
  }
}