Primitive permutation groups and derangements of prime power order

Tim C Burness, Hung Tong-Viet

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

Let G be a transitive permutation group on a finite set of size at least 2. By a well known theorem of Fein, Kantor and Schacher, G contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an r-power, for some fixed prime r. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group G has this property if and only if every two-point stabilizer is an r-group. Here the structure of G has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on r′-semiregular pairs.
Original languageEnglish
Pages (from-to)255-291
Number of pages37
JournalManuscripta Mathematica
Volume150
Issue number1
Early online date6 Nov 2015
DOIs
Publication statusPublished - May 2016

Keywords

  • Primary 20B15
  • secondary 20D05

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