ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 05c25 Li, Cai Heng, Mathematics Dept. UWA, AUS-6907 Nedlands, WA 6907, Australia On Isomorphisms of Finite Cayley Graphs For a finite group G and a self-inverse subset S ⊂ G \ {1}, the associated Cayley graph is the graph Cay(G, S) with vertex set G and with x adjacent to y iff yx^{−1} ∈ S. Clearly each σ ∈ Aut(G) induces an isomorphism from Cay(G, S) to Cay(G, S^{σ}). A group G is called a CI-group if, for all S, T ⊂ G, Cay(G, S) ≅ Cay(G, T) iff S^{σ} = T for some σ ∈ Aut(G). The problem of seeking finite CI-groups is a long-standing open problem in the area of Cayley graphs. Here we show that a CI-group G has form G = U ×V, where (|U|, |V |) = 1, U is abelian with all Sylow subgroups elementary abelian, and V belongs to an explicitly determined list of groups. Then we discuss the problem of completely classifying finite CI-groups.