Finite -arc transitive Cayley graphs and flag-transitive projective planes

Author:
Cai Heng Li

Journal:
Proc. Amer. Math. Soc. **133** (2005), 31-41

MSC (2000):
Primary 20B15, 20B30, 05C25

DOI:
https://doi.org/10.1090/S0002-9939-04-07549-5

Published electronically:
July 26, 2004

MathSciNet review:
2085150

Full-text PDF

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Abstract: In this paper, a characterisation is given of finite -arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of -transitive Cayley graphs with such that all -transitive Cayley graphs of valency are their normal covers. This indicates that -arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

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Additional Information

**Cai Heng Li**

Affiliation:
School of Mathematics and Statistics, The University of Western Australia, Crawley, 6009 Western Australia, Australia

Email:
li@maths.uwa.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-04-07549-5

Received by editor(s):
August 27, 2003

Received by editor(s) in revised form:
September 11, 2003, and September 24, 2003

Published electronically:
July 26, 2004

Additional Notes:
This work was supported by an Australian Research Council Discovery Grant, and a QEII Fellowship. The author is grateful to the referee for his constructive comments.

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2004
American Mathematical Society