Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups

TL;DR

This paper classifies factorizations of almost simple groups with solvable factors and applies this to characterize certain highly symmetric Cayley graphs of solvable groups, revealing that most such graphs are covers of well-known graphs.

Contribution

It provides a complete classification of group factorizations involving solvable factors and characterizes 3-arc-transitive Cayley graphs of solvable groups.

Findings

Most non-bipartite 3-arc-transitive Cayley graphs of solvable groups are covers of Petersen or Hoffman-Singleton graphs.

A comprehensive classification of factorizations of almost simple groups with solvable factors.

Identification of the structure of highly symmetric Cayley graphs of solvable groups.

Abstract

A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the cycles, every non-bipartite connected 3-arc-transitive Cayley graph of a solvable group is a cover of the Petersen graph or the Hoffman-Singleton graph.