Two-Arc Transitive Graphs and Twisted Wreath Products
Robert W. Baddeley
DOI: 10.1023/A:1022447514654
Abstract
The paper addresses a part of the problem of classifying all 2-arc transitive graphs: namely, that of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices. A construction is given for such groups, together with the associated graphs, in terms of the following ingredients: a nonabelian simple group T, a permutation group P acting 2-transitively on a set Ĩ \in F( 
) and such that Tis generated by F( ). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.
) and such that Tis generated by F( ). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.Pages: 215–237
Keywords: graph; group; two-arc transitive graph; wreath product
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References
1. R.W. Baddeley, "Primitive permutation groups of twisted wreath type with a symmetric group as a point-stabilizer," in preparation.
2. R.W. Baddeley, "Primitive permutation groups with a regular non-abelian normal subgroup," Proc. London Math. Soc. to appear.
3. R. Bercov, "On groups without abelian composition factors," J. Algebra 5 (1967), 106-109.
4. P.J. Cameron, "Finite permutation groups and finite simple groups," Bull. London Math. Soc. 13 (1981), 1-22.
5. R.T. Curtis, "Natural constructions of the Mathieu groups," Math. Proc. Cam. Phil. Soc. 106 (1989), 423-429.
6. P. Forster and L.G. Kovacs, "Finite primitive groups with a single non-abelian regular normal subgroup," Research Report 17, ANU-MSRC, 1989.
7. W. Kimmerle, R. Lyons, R. Sandling, and D.N. Teague, "Composition factors from the group ring and Artin's theorem on orders of simple groups," Proc. London Math. Soc. 60 (1990), 89-122.
8. J. Lafuente, "On restricted twisted wreath products of groups," Arch. Math. 43 (1984), 208-209.
9. M.W. Liebeck, C.E. Praeger, and J. Saxl, "A classification of the maximal subgroups of the finite alternating and symmetric groups," J. Algebra 111 (1987), 365-383.
10. B.H. Neumann, "Twisted wreath products of groups," Arch. Math. 14 (1963), 1-6.
11. C.E. Praeger, "Finite vertex transitive graphs and primitive permutation groups," Proceedings Marshall Hall Conference, (Vermont, 1990), to appear.
12. C.E. Praeger, "The inclusion problem for finite primitive permutation groups," Proc. London Math. Soc. 60 (1990), 68-88.
13. C.E. Praeger, "An O'Nan-Scott theorem for finite quasiprimitive permutation groups, and an application to 2-arc transitive graphs," J. London Math. Soc. to appear.
14. C.E. Praeger, "Primitive permutation groups with a doubly transitive subconstituent," J. Australian Math. Soc. 45 (1988), 66-77.
15. C.E. Praeger, "On a reduction theorem for finite bipartite 2-arc transitive graphs," Australas. J. Combin. to appear.
16. M. Suzuki, Group Theory 1, Springer, 1982.
17. R. Weiss, "The non-existence of 8-transitive graphs," Combinatorica 1 (1981), 309-311.
2. R.W. Baddeley, "Primitive permutation groups with a regular non-abelian normal subgroup," Proc. London Math. Soc. to appear.
3. R. Bercov, "On groups without abelian composition factors," J. Algebra 5 (1967), 106-109.
4. P.J. Cameron, "Finite permutation groups and finite simple groups," Bull. London Math. Soc. 13 (1981), 1-22.
5. R.T. Curtis, "Natural constructions of the Mathieu groups," Math. Proc. Cam. Phil. Soc. 106 (1989), 423-429.
6. P. Forster and L.G. Kovacs, "Finite primitive groups with a single non-abelian regular normal subgroup," Research Report 17, ANU-MSRC, 1989.
7. W. Kimmerle, R. Lyons, R. Sandling, and D.N. Teague, "Composition factors from the group ring and Artin's theorem on orders of simple groups," Proc. London Math. Soc. 60 (1990), 89-122.
8. J. Lafuente, "On restricted twisted wreath products of groups," Arch. Math. 43 (1984), 208-209.
9. M.W. Liebeck, C.E. Praeger, and J. Saxl, "A classification of the maximal subgroups of the finite alternating and symmetric groups," J. Algebra 111 (1987), 365-383.
10. B.H. Neumann, "Twisted wreath products of groups," Arch. Math. 14 (1963), 1-6.
11. C.E. Praeger, "Finite vertex transitive graphs and primitive permutation groups," Proceedings Marshall Hall Conference, (Vermont, 1990), to appear.
12. C.E. Praeger, "The inclusion problem for finite primitive permutation groups," Proc. London Math. Soc. 60 (1990), 68-88.
13. C.E. Praeger, "An O'Nan-Scott theorem for finite quasiprimitive permutation groups, and an application to 2-arc transitive graphs," J. London Math. Soc. to appear.
14. C.E. Praeger, "Primitive permutation groups with a doubly transitive subconstituent," J. Australian Math. Soc. 45 (1988), 66-77.
15. C.E. Praeger, "On a reduction theorem for finite bipartite 2-arc transitive graphs," Australas. J. Combin. to appear.
16. M. Suzuki, Group Theory 1, Springer, 1982.
17. R. Weiss, "The non-existence of 8-transitive graphs," Combinatorica 1 (1981), 309-311.