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Isomorphisms of Cayley digraphs of Abelian groups

Published online by Cambridge University Press:  17 April 2009

Cai Heng Li
Cai Heng Li
Affiliation:
Department of MathematicsUniversity of Western AustraliaNedlands WA 6907Australia e-mail: li@maths.uwa.edu.au
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Abstract

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For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 181 - 188
Copyright
Copyright © Australian Mathematical Society 1998

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