Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants
Cai Heng Li
Department of Mathematics Yunnan University Kunming 650031 People's Republic of China
DOI: 10.1007/s10801-005-6903-3
Abstract
A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970 S[[ `( K)] b] Σ[{\bar K}_b] , a deleted lexicographic product S[[ `( K)] b] - b S Σ[{\bar K}_b] - bΣ , where 
is a smaller arc transitive circulant, or 
is a normal circulant, that is, Auta has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.
is a smaller arc transitive circulant, or is a normal circulant, that is, Auta has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.Pages: 131–136
Keywords: keywords cyclic regular subgroup; arc transitive circulant
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References
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14. C.H. Li, “Edge-transitive Cayley graphs and rotary Cayley maps,” Trans. Amer. Math. Soc. (to appear).
15. C.H. Li, D. Maru\check si\check c, and J. Morris, “A classification of circulant arc-transitive graphs of square-free order,” J. Algebraic Combin. 14 (2001), 145-151.
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17. H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
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19. M.Y. Xu, H.S. Sim, and Y.J. Baik, “Arc-transitive circulant digraphs of odd prime-power order,” Discrete Math. 287 (2004), 113-119.
2. C.Y. Chao, “On the classification of symmetric graphs with a prime number of vertices,” Trans. Amer. Math. Soc. 158 (1971), 247-256.
3. C.Y. Chao and J.G. Wells, “A class of vertex-transitive digraphs,” J. Combin. Theory Ser. B 14 (1973), 246-255.
4. S.A. Evdokimov and I.N. Ponomarenko, “Characterization of cyclotomic schemes and normal Schur rings over a cyclic group, (Russian),” Algebra i Analiz 14(2) (2002), 11-55.
5. W. Feit, “Some consequences of the classification of finite simple groups,” The Santa Cruz Conference on Finite Groups Univ. California, Santa Cruz, Calif., 1979, pp. 175-181, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.
6. C.D. Godsil, “On the full automorphism group of a graph,” Combinatorica 1 (1981), 243-256.
7. D. Gorenstein, Finite Simple Groups, Plenum Press, New York, 1982.
8. G. Jones, “Cyclic regular subgroups of primitive permutation groups,” J. Group Theory 5(4) (2002), 403-407.
9. W. Kantor, Some consequences of the classification of finite simple groups, in Finite Groups-Coming of Age, John McKay (Ed.), Amer. Math. Soc., 1982.
10. I. Kovács, “Classifying arc-transitive circulants,” J. Algebraic Combin. 20 (2004), 353-358.
11. K.H. Leung and S.H. Man, “On Schur rings over cyclic groups,” II, J. Algebra 183 (1996), 173-285.
12. K.H. Leung and S. H. Man, “On Schur rings over cyclic groups,” Israel J. Math. 106 (1998), 251-267.
13. C.H. Li, “The finite primitive permutation groups containing an abelian regular subgroup,” Proc. London Math. Soc. 87 (2003), 725-748.
14. C.H. Li, “Edge-transitive Cayley graphs and rotary Cayley maps,” Trans. Amer. Math. Soc. (to appear).
15. C.H. Li, D. Maru\check si\check c, and J. Morris, “A classification of circulant arc-transitive graphs of square-free order,” J. Algebraic Combin. 14 (2001), 145-151.
16. J.X. Meng and J.Z. Wang, “A classification of 2-arc-transitive circulant digraphs,” Disc. Math. 222 (2000), 281-284.
17. H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
18. H. Wielandt, Permutation groups through invariant relations and invariant functions, in Mathematische Werke/Mathematical works. Vol.
1. Group theory, Bertram Huppert and Hans Schneider (Eds.), Walter de Gruyter Co., Berlin, 1994, pp. 237-266.
19. M.Y. Xu, H.S. Sim, and Y.J. Baik, “Arc-transitive circulant digraphs of odd prime-power order,” Discrete Math. 287 (2004), 113-119.