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Pentavalent 1-transitive digraphs with non-solvable automorphism groups. (English) Zbl 07344163
Summary: A digraph \(\overrightarrow{\Gamma}\) is said to be 1-transitive if its automorphism group acts transitively on the 1-arcs but not on the 2-arcs of \(\overrightarrow{\Gamma}\). We give a tentatively complete classification of pentavalent strongly connected 1-transitive digraphs of order \(2^ap^bq\), where \(p\) and \(q\) are two distinct odd primes, \(a\in\{3,\dots,8\}\), \(b\in\{1,\dots,4\}\), whose automorphism groups are non-solvable. It is shown that such digraphs exist if and only if \(q=3\) or 13 and \(p\in\{7,11, 17,19,31,41\}\).
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C20 Directed graphs (digraphs), tournaments
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Full Text: DOI
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