On Cayley graphs of abelian groups (Q1272901)

The author proves several results concerning Cayley graphs of finite abelian groups. Let \(A_1\) be the stabilizer of the vertex \(1\) in the automorphism group of the Cayley graph of \(G\) with respect to a subset \(1 \not\in S \subseteq G\). The author proves that if \(A_1\) acts unfaithfully on \(S\), then \(S\) contains a coset of some nontrivial subgroup of \(G\). This result is applied to the following question: for an integer \(m>0\), a group \(G\) is called an \(m\)-DCI-group if whenever the Cayley graphs \(\text{C}(G,S)\) and \(\text{C}(G,T)\) are isomorphic where \(S,T\subseteq G\) and \(| S| \leq m\), then \(S = T^{\alpha}\) for some \(\alpha \in \text{Aut}(G)\). The author determines all \(m\)-DCI \(p\)-groups for prime \(p\) and \(2\leq m\leq p+1\).