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FINITE NORMAL 2-GEODESIC TRANSITIVE CAYLEY GRAPHS

Part of: Permutation groups Algebraic combinatorics

Published online by Cambridge University Press:  16 March 2016

WEI JIN
WEI JIN*
Affiliation:
School of Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, PR China Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, PR China email [email protected]
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Abstract

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For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

Keywords

2-geodesic transitive graphCayley graphautomorphism group

MSC classification

Primary: 05E18: Group actions on combinatorial structures
Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 3 , June 2016 , pp. 338 - 348
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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