The Two-Arc-Transitive Graphs of Square-Free Order Admitting Alternating or Symmetric Groups

Abstract

Let G be a finite group with soc(G) = A c for c ≥ 5. A characterization of the subgroups with square-free index in G is given. Also, it is shown that a (G, 2)-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of 11 other graphs.2010 Mathematics subject classification: primary 20B15; secondary 20B30.

Generalised Voltage Graphs

“…If G acts transitively on the arcs (ordered pairs of adjacent vertices) of Γ, then Γ can be reconstructed as a coset-graph from the group G, the vertex-stabiliser G v and an element a ∈ G where a is an element swapping the vertex v with a neighbour of v; see [2, Section 1.2] or [22], for example. Describing arc-transitive graphs as coset graphs is a standard method often used in the classification results (see [3,5,10,23,27], to name a few) as well as a way to store a graph in a database (see [1]).…”