Normal minimal Cayley digraphs of abelian groups (Q1568790)

Let \(G\) be a finite abelian group (additively written) and \(0\) be the neutral element of \(G\). For any subset \(S\) of \(G-\{0\}\), we denote by \(C(G,S)\) the Cayley digraph assigned to the pair \(G\), \(S\). We say that \(C(G,S)\) is minimal if \(S\) is a minimal generating system of \(G\); moreover, \(C(G,S)\) is called normal if the left regular representation of \(G\) is a normal subgroup of the automorphism group of \(C(G,S)\). The authors are interested in the normality of minimal Cayley graphs. Let a minimal generating system \(S\) of \(G\) be considered. Introduce a partition of \(S\) (into the classes \(S_1,S_2,\dots, S_k\)) by the rule that two elements \(s_\alpha\) and \(s_\beta\) are in a common class precisely when \(2s_\alpha= 2s_\beta\). For any \(S_i\), denote by \(\lambda(S_i)\) the smallest number \(q\) such that the implication \(s_i\in S_i\Rightarrow qs_i\in S_i\) is not valid. The main result of the paper asserts that \(C(G,S)\) is not normal exactly when \(|S_i|= 2< \lambda(S_i)\) holds for some \(i\) with \(1\leq i\leq k\). In another theorem, the groups for which each minimal Cayley graph is normal are characterized in terms of the non-existence of direct summands of type \(Z_2\oplus Z_{2^m}\).