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The exponent of the primitive Cayley digraphs on finite Abelian groups

✍ Scribed by Wang Jian-Zhong; Meng Ji-Xiang

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
811 KB
Volume
80
Category
Article
ISSN
0166-218X

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✦ Synopsis

Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph Xc~,s) to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X 1, (6 S) on an Abelian group is primitive if and only if S-'S is a generating set for G. Moreover, it is shown that if a Cayley digraph X(G,S) on an Abelian group is primitive, then its exponent either is n -1, [:I, [I] -1 or is not exceeding [ 51 + 1. Finally, we also characterize those Cayley digraphs on Abelian groups with exponent n -1, [;I, [I] -1. In particular, we generalize a number of well-known results for the primitive circulant matrices.

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