Isomorphisms of finite Cayley digraphs of bounded valency (Q1127875)

For a finite group \(G\) and a subset \(S\) of \(G\setminus \{1\}\), let \(\text{Cay} (G,S)\) denote the Cayley digraph of \(G\) with respect to \(S\), that is, the directed graph with vertex set \(G\) and edge set \(\{(a,b)\mid a,b\in G\), \(ba^{-1}\in S\}\). If \(\sigma\in\Aut (G)\) then \(\sigma\) induces an isomorphism from \(\text{Cay}(G,S)\) to \(\text{Cay} (G,S^\sigma)\). If the converse holds, that is, for all \(T\), \(\text{Cay} (G,S)\cong \text{Cay} (G,T)\) implies that \(S^\sigma= T\) for some \(\sigma\in \Aut(G)\), then \(\text{Cay} (G,S)\) is called a CI-graph. This paper investigates, for integers \(m\geq 2\), finite groups \(G\) for which all Cayley digraphs of valency at most \(m\) are CI-graphs. It is shown that such groups \(G\) have the form \(U\times V\), where \((| U|,| V|)=1\), \(U\) is abelian and \(V\) belongs to an explicitly determined list of groups. Moreover Sylow subgroups of such groups satisfy some very restrictive conditions.