TY - JOUR

T1 - On the isomorphism problem for finite Cayley graphs of bounded valency

AU - Li, Cai-Heng

AU - Praeger, Cheryl

PY - 1999

Y1 - 1999

N2 - For a subset S of a group G such that 1 is not an element of S and S = S-1 , the associated Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx(-1) epsilon S. Each sigma epsilon Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, S-sigma). For a positive integer m, the group G is called an m-CI-group if, for all Cayley subsets S of size;at most m, whenever Cay(G, S) congruent to Cay(G, T) there is an element sigma epsilon Aut(G) such that S-sigma = T. It is shown that if G is an m-CI-group for some m greater than or equal to 4, then G = U x V, where (\U\, \V\) = 1, U is abelian, and V belongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification. (C) 1999 Academic Press.

AB - For a subset S of a group G such that 1 is not an element of S and S = S-1 , the associated Cayley graph Cay(G, S) is the graph with vertex set G such that {x, y} is an edge if and only if yx(-1) epsilon S. Each sigma epsilon Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, S-sigma). For a positive integer m, the group G is called an m-CI-group if, for all Cayley subsets S of size;at most m, whenever Cay(G, S) congruent to Cay(G, T) there is an element sigma epsilon Aut(G) such that S-sigma = T. It is shown that if G is an m-CI-group for some m greater than or equal to 4, then G = U x V, where (\U\, \V\) = 1, U is abelian, and V belongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification. (C) 1999 Academic Press.

U2 - 10.1006/eujc.1998.0291

DO - 10.1006/eujc.1998.0291

M3 - Article

VL - 20

SP - 279

EP - 292

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

ER -