The case of equality in the Livingstone-Wagner Theorem
David Bundy1
and Sarah Hart2
1Universität zu Kiel Mathematisches Seminar Ludewig-Meyn Straße 4 Kiel 24098 Germany
2Birkbeck, University of London School of Economics, Mathematics and Statistics Malet Street London WC1E 7HX UK
2Birkbeck, University of London School of Economics, Mathematics and Statistics Malet Street London WC1E 7HX UK
DOI: 10.1007/s10801-008-0130-7
Abstract
Let G be a permutation group acting on a set Ω of size n\in \Bbb N and let 1\leq k<( n - 1)/2. Livingstone and Wagner proved that the number of orbits of G on k-subsets of Ω is less than or equal to the number of orbits on ( k+1)-subsets. We investigate the cases when equality occurs.
Pages: 215–227
Keywords: keywords livingstone-wagner theorem; permutation groups; orbits; partitions
Full Text: PDF
References
1. Cameron, P.J.: Transitivity of permutation groups on unordered sets. Math. Z. 148(2), 127-139 (1976)
2. Cameron, P.J.: Orbits of permutation groups on unordered sets. J. London Math. Soc. (2) 17(3), 410- 414 (1978)
3. Cameron, P.J.: Orbits of permutation groups on unordered sets. II. J. London Math. Soc. (2) 23(2), 249-264 (1981)
4. Cameron, P.J., Neumann, P.M., Saxl, J.: An interchange property in finite permutation groups. Bull. London Math. Soc. 11(2), 161-169 (1979)
5. Cameron, P.J., Thomas, S.: Groups acting on unordered sets. Proc. London Math. Soc. (3) 59(3), 541- 557 (1989)
6. GAP-Groups, Algorithms and Programming, Version 4.4.4 (2004).
7. Livingstone, D., Wagner, A.: Transitivity of finite permutation groups on unordered sets. Math. Z. 90, 393-403 (1965)
8. Robinson, G. de B.: Note on a theorem of Livingstone and Wagner. Math. Z. 102, 351-352 (1967)
9. Schur, I.: Vorlesungen Über Invariantentheorie. Bearbeitet und herausgegeben von Helmut Grunsky.
2. Cameron, P.J.: Orbits of permutation groups on unordered sets. J. London Math. Soc. (2) 17(3), 410- 414 (1978)
3. Cameron, P.J.: Orbits of permutation groups on unordered sets. II. J. London Math. Soc. (2) 23(2), 249-264 (1981)
4. Cameron, P.J., Neumann, P.M., Saxl, J.: An interchange property in finite permutation groups. Bull. London Math. Soc. 11(2), 161-169 (1979)
5. Cameron, P.J., Thomas, S.: Groups acting on unordered sets. Proc. London Math. Soc. (3) 59(3), 541- 557 (1989)
6. GAP-Groups, Algorithms and Programming, Version 4.4.4 (2004).
7. Livingstone, D., Wagner, A.: Transitivity of finite permutation groups on unordered sets. Math. Z. 90, 393-403 (1965)
8. Robinson, G. de B.: Note on a theorem of Livingstone and Wagner. Math. Z. 102, 351-352 (1967)
9. Schur, I.: Vorlesungen Über Invariantentheorie. Bearbeitet und herausgegeben von Helmut Grunsky.