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On the Isomorphism Problem for Finite Cayley Graphs of Bounded Valency

✍ Scribed by C.H. Li; C.E. Praeger

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
187 KB
Volume
20
Category
Article
ISSN
0195-6698

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✦ Synopsis

For a subset S of a group G such that 1 / ∈ 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 ∈ S. Each Οƒ ∈ Aut(G) induces an isomorphism from Cay(G, S) to the Cayley graph Cay(G, S Οƒ ). 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)

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.

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Let G be a finite group, S a subset of G=f1g; and let Cay ðG; SÞ denote the Cayley digraph of G with respect to S: If, for any subset T of G=f1g; CayðG; SÞ ffi CayðG; T Þ implies that S a ¼ T for some a 2 AutðGÞ; then S is called a CI-subset. The group G is called a CIM-group if for any minimal gene

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