The Abstract

	Annals of Combinatorics 2 (1998) 85-101 Proof of a Conjecture on the Sperner Property of the Subgroup Lattice of an Abelian p-Group Jun Wang Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China junwang@dlut.edu.cn Received March 15, 1997 AMS Subject Classification: 05A05, 05D05,06A07 Abstract. Let L_(kⁿ)(p) denote the subgroup lattice of the abelian p-group (Z/p^kZ) × ··· × (Z/p^kZ) (n times). It is conjectured that the lattice has the Sperner property. When k=1, the conjecture is true since it is isomorphic to the subspace lattice, and Stanley has confirmed it for k=2. In this paper, we prove that the conjecture is generally true. Keywords: Sperner property, poset, subgroup lattice References 1. M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979. 2. I. Anderson, Combinatorics of Finite Sets, Oxford University Press, Oxford, 1987. 3. G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38 (2) (1934–35) 385–401. 4. L.M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 (1987) 771–775. 5. E.R. Canfield, On a problem of Rota, Adv. Math. 29 (1978) 1–10. 6. K. Engel and H.-D. Gronau, Sperner Theory in Partially Ordered Sets, Teubner, Leipzig, 1985. 7. K. Engel, Sperner Theory, Cambridge University Press, Cambridge, 1997. 8. R. Freese, An application of Dilworth’s lattice of maximal antichains, Discrete Math. 7 (1974) 107–109. 9. C. Greene and D.J. Kleitman, The structure of Sperner k-families, J. Combin. Theory Ser. A 20 (1976) 41–68. 10. C. Greene and D.J. Kleitman, Strong versions of Sperner’s theorem, J. Combin. Theory Ser. A 20 (1976) 80–88. 11. C. Greene and D.J. Kleitman, Proof techniques in the ordered sets, In: Studies in Combinatorics, G.-C. Rota, Ed., Math. Assn. America, Washington DC, 1978, pp. 22–79. 12. J.R. Griggs, On chains and Sperner k-families in ranked posets, J. Combin. Theory Ser. A 28 (1980) 156–168. 13. J.R. Griggs, M. Saks, and D. Sturtevant, On chains and Sperner k-families in ranked posets II, J. Combin. Theory Ser. A 29 (1980) 391–394. 14. J.R. Griggs, Matchings, cutsets, and chain partitions in graded posets, Discrete Math. 144 (1995) 33–46. 15. D. Kleitman, M. Edelberg, and D. Lubell, Maximal-sized antichains in partial orders, Discrete Math. 1 (1971) 47–53. 16. K.W. Lih, Sperner families over a subset, J. Combin. Theory Ser. A 29 (1980) 182–185. 17. D. Lubell, A short proof of Sperner’s theorem, J. Combin. Theory 1 (1966) 299. 18. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995. 19. G.-C. Rota, Research problem 2-1, J. Combin. Theory 2 (1967) 104. 20. G.-C. Rota and L.H. Harper, Matching theory, an introduction, In: Advances in Probability Vol. 1, Marcel Dekker, Inc., New York, 1971. 21. E. Sperner, Ein Satzüber Untermengen einer endlichen Menge, Math. Z. 27 (1928) 327–330. 22. R.P. Stanley, Enumerative Combinatorics Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986. 23. R.P. Stanley, Weyl groups, the hard Lefschets theorem and the Sperner property, SIAM J. Alg. Disc. Meth. 1 (1980) 168–184. 24. R.P. Stanley, Some applications of algebra to combinatorics, Discrete Appl. Math. 34 (1991) 241–277. Get the LaTex \| DVI \| PS file of this abstract. back

Annals of Combinatorics 2 (1998) 85-101

Proof of a Conjecture on the Sperner Property of the Subgroup Lattice of an Abelian p-Group

Jun Wang

Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
junwang@dlut.edu.cn

Received March 15, 1997

AMS Subject Classification: 05A05, 05D05,06A07

Abstract. Let L_(kⁿ)(p) denote the subgroup lattice of the abelian p-group

(Z/p^kZ) × ··· × (Z/p^kZ) (n times). It is conjectured that the lattice has the Sperner property. When k=1, the conjecture is true since it is isomorphic to the subspace lattice, and Stanley has confirmed it for k=2. In this paper, we prove that the conjecture is generally true.

Keywords: Sperner property, poset, subgroup lattice

References

1. M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979.

2. I. Anderson, Combinatorics of Finite Sets, Oxford University Press, Oxford, 1987.

3. G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38 (2) (1934–35) 385–401.

4. L.M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 (1987) 771–775.

5. E.R. Canfield, On a problem of Rota, Adv. Math. 29 (1978) 1–10.

6. K. Engel and H.-D. Gronau, Sperner Theory in Partially Ordered Sets, Teubner, Leipzig, 1985.

7. K. Engel, Sperner Theory, Cambridge University Press, Cambridge, 1997.

8. R. Freese, An application of Dilworth’s lattice of maximal antichains, Discrete Math. 7 (1974) 107–109.

9. C. Greene and D.J. Kleitman, The structure of Sperner k-families, J. Combin. Theory Ser. A 20 (1976) 41–68.

10. C. Greene and D.J. Kleitman, Strong versions of Sperner’s theorem, J. Combin. Theory Ser. A 20 (1976) 80–88.

11. C. Greene and D.J. Kleitman, Proof techniques in the ordered sets, In: Studies in Combinatorics, G.-C. Rota, Ed., Math. Assn. America, Washington DC, 1978, pp. 22–79.

12. J.R. Griggs, On chains and Sperner k-families in ranked posets, J. Combin. Theory Ser. A 28 (1980) 156–168.

13. J.R. Griggs, M. Saks, and D. Sturtevant, On chains and Sperner k-families in ranked posets II, J. Combin. Theory Ser. A 29 (1980) 391–394.

14. J.R. Griggs, Matchings, cutsets, and chain partitions in graded posets, Discrete Math. 144 (1995) 33–46.

15. D. Kleitman, M. Edelberg, and D. Lubell, Maximal-sized antichains in partial orders, Discrete Math. 1 (1971) 47–53.

16. K.W. Lih, Sperner families over a subset, J. Combin. Theory Ser. A 29 (1980) 182–185.

17. D. Lubell, A short proof of Sperner’s theorem, J. Combin. Theory 1 (1966) 299.

18. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Second edition, 1995.

19. G.-C. Rota, Research problem 2-1, J. Combin. Theory 2 (1967) 104.

20. G.-C. Rota and L.H. Harper, Matching theory, an introduction, In: Advances in Probability Vol. 1, Marcel Dekker, Inc., New York, 1971.

21. E. Sperner, Ein Satzüber Untermengen einer endlichen Menge, Math. Z. 27 (1928) 327–330.

22. R.P. Stanley, Enumerative Combinatorics Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986.

23. R.P. Stanley, Weyl groups, the hard Lefschets theorem and the Sperner property, SIAM J. Alg. Disc. Meth. 1 (1980) 168–184.

24. R.P. Stanley, Some applications of algebra to combinatorics, Discrete Appl. Math. 34 (1991) 241–277.

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