Further restrictions on the structure of finite CI-groups

Li, Cai Heng ; Lu, Zai Ping ; Pálfy, P.P.

Journal of Algebraic Combinatorics, Tome 26 (2007) no. 2, pp. 161-181.

Résumé

Summary: A group $G$ is called a $CI-group$ if, for any subsets $S, T\subset G$, whenever two Cayley graphs $Cay( G, S)$ and $Cay( G, T)$ are isomorphic, there exists an element $\sigma \epsilon Aut( G)$ such that $S\sigma = T$. The problem of seeking finite CI-groups is a long-standing open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius groups of order $4 p$ and $6 p$, and the metacyclic groups of order $9 p$ of which the centre has order 3 are not CI-groups, where $p$ is an odd prime. Then a shorter explicit list is given of candidates for finite CI-groups. Finally, some new families of finite CI-groups are found, that is, the metacyclic groups of order $4 p$ (with centre of order 2) and of order $8 p$ (with centre of order 4) are CI-groups, and a proof is given for the Frobenius group of order $3 p$ to be a CI-group, where $p$ is a prime.

Détail

Classification : One, of, the, purposes, of, this, paper, is, to, give, an, improvement, of, the, description, of, finite, CI-groups, obtained, in, [21],, and, moreover, the, argument, used, in, the, paper, is, independent, of, the, classification, of, finite, simple, groups
Keywords: keywords Cayley graphs, isomorphism problem, CI-groups