×

On the normality of Cayley digraphs of groups of order twice a prime. (English) Zbl 0917.05037

Given an arbitrary Cayley (di)graph \(\Gamma=C(G,X)\), the general problem of determining the full automorphism group of \(\Gamma\), \(\operatorname{Aut} \Gamma\), appears to be rather complicated. Despite the fact that we know that \(\operatorname{Aut}\Gamma\) contains the underlying group \(G\) together with all group automorphisms preserving the generating set \(X\), all the known methods for determining the rest of \(\operatorname{Aut}\Gamma\) involve computations that do not take a particular advantage of the group structure of \(\Gamma\).
The paper in consideration deals with Cayley (di)graphs that do not admit additional automorphisms at all, i.e., the paper is devoted to Cayley graphs \(\Gamma= C(G, X)\) whose full automorphism group \(\operatorname{Aut}\Gamma\) is a semidirect extension of \(G\) by the group of automorphisms of \(G\) preserving \(X\). Cayley (di)graphs satisfying this property have been named normal Cayley (di)graphs by \(Xu\).
All Cayley (di)graphs of prime order \(p\) different from the complete graph and its complement are known to be normal. The authors consider the next class of Cayley (di)graphs—graphs of order \(2p\), \(p\) a prime, and classify all the non-normal Cayley (di)graphs of this order. The results are useful and interesting and were obtained via careful investigation of all possible cases together with a detailed knowledge of permutation groups and their actions of degree \(2p\).

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
PDFBibTeX XMLCite