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Graphical Frobenius representations

Authors
  • Doyle, John Kevin1
  • Tucker, Thomas W.2
  • Watkins, Mark E.3
  • 1 Benedictine University, College of Business, Lisle, IL, USA , Lisle (United States)
  • 2 Colgate University, Department of Mathematics, Hamilton, NY, USA , Hamilton (United States)
  • 3 Syracuse University, Department of Mathematics, Syracuse, NY, USA , Syracuse (United States)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Feb 06, 2018
Volume
48
Issue
3
Pages
405–428
Identifiers
DOI: 10.1007/s10801-018-0814-6
Source
Springer Nature
Keywords
License
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Abstract

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A graphical Frobenius representation (GFR) of a Frobenius group G is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to G. The problem of classifying which Frobenius groups admit a GFR is a natural extension of the classification of groups that have a graphical regular representation (GRR), which occupied many authors from 1958 through 1982. In this paper, we review for graph theorists some standard and deep results about finite Frobenius groups, determine classes of finite Frobenius groups and individual groups that do and do not admit GFRs, and classify those Frobenius groups of order at most 300 having a GFR. Because a Frobenius group, as opposed to a regular permutation group, has a highly restricted structure, the GFR problem emerges as algebraically more complex than the GRR problem. This paper concludes with some further questions and a strong conjecture.

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