On locally primitive Cayley graphs of finite simple groups |
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Authors: | Xingui Fang |
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Affiliation: | a LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, PR China b Department of Mathematics, Capital Normal University, Beijing 100048, PR China |
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Abstract: | In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal. |
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Keywords: | Finite simple group Maximal subgroup Cayley graph Normal Cayley graph Arc-transitive graph Locally primitive graph |
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