{
  "id": "0803.4175",
  "version": "v2",
  "published": "2008-03-28T17:45:05.000Z",
  "updated": "2008-10-01T14:54:57.000Z",
  "title": "Parity patterns associated with lifts of Hecke groups",
  "authors": [
    "Christian Krattenthaler",
    "Thomas W. Müller"
  ],
  "comment": "AmS-LaTeX; 49 pages; minor corrections, Section 5 restructured for better reading (with some proofs put in a new appendix), new numbering of theorems",
  "journal": "Abh. Math. Sem. Univ. Hamburg 78 (2008), 99-147.",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let $q$ be an odd prime, $m$ a positive integer, and let $\\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\\Ga_m(q)$ which were defined in [T. W. M\\\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\\Ga_m(q)$ into a given finite group $H$, in the case when $\\gcd(m,| H|)=1$. This computation depends upon the solution of three counting problems in the Hecke group $\\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\\\"uller, in: {\\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\\\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-10-01T14:54:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E06",
      "05A15",
      "05E99",
      "16W22",
      "20E07"
    ],
    "keywords": [
      "hecke group",
      "parity patterns",
      "free product",
      "lms lecture notes series",
      "subgroup numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.4175K"
    }
  }
}