On the classes of Steiner loops of small orders

M. H. Armanious
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 17, 2011, Number 2, Pages 52–68
Full paper (PDF, 565 Kb)

Details

Authors and affiliations

M. H. Armanious
Mathematics Department, Faculty of Science, Mansoura University
Mansoura, Egypt

Abstract

According to the number of sub-SL(8)s (sub-STS(7)s), there are five classes of sloops SL(16)s (STS(15)s) [2, 5].) In [4] the author has classified SL(20)s into 11 classes. Using computer technique in [10] the authors gave a large number for each class of SL(20)s. There are only simple SL(22)s and simple SL(26)s. So the next admissible cardinality is 28. Also, all SL(32)s are classified into 14 classes in [3]. We try to generalize the classification of SL(20)s given in [4] for SL(2n)s for each possible n and applying this method for n = 14 to classify all possible classes of SL(28)s. Consequently, we can establish all classes of nonsimple SL(28)s and all classes of semi-planar SL(28)s (STS(27)s). In this article, we show in section 3 that there are nine classes of SL(28)s (STS(27)s having one sub-SL(14) (sub-STS(13)) and r sub-SL(8)s (sub-STS(7)s) for r = 0, 1, 2, 3, 4, 5, 8, 11 or 16. All these sloops are subdirectly irreducible having exactly one proper homomorphic image isomorphic to SL(2). In section 4, we construct all classes of semi-planar SL(28)s. Such SL(28)s (STS(27)s) have r sub-SL(8)s (sub-STS(7)s) for r = 1, 2, 3, 4, 5, 8, 11, 16 but no sub-SL(14) (sub-STS(13)).

In [4] is determined a necessary and sufficient condition for a sub-SL(2) = {1, x} of an SL(2n) to be normal. This result supplies us with the following two facts. First, there is another nonsimple subdirectly irreducible SL(2n) having exactly one proper homomorphic image isomorphic to an SL(n). Accordingly, we can construct all classes of nonsimple subdirectly irreducible SL(28)s. Second fact is that if an SL(2n) has a simple sub-SL(n) and (n – 1)(n – 2)/6 sub-SL(8)s passing through a non-unit element, then SL(2n) is isomorphic to the direct product SL(n) × SL(2). According to the result of section 3 and the above two facts, we may say that there are 8 simple classes of SL(28)s and only 11 classes of nonsimple SL(28)s, all these classes have no sub-SL(10)s. In the last section, we construct an example for each class given above of nonmsimple and simple (semi-planar) SL(28)s (STS(27)s). Finally, we review the classes of SL(2n)s (STS(2n – 1)s) in 3 tables for 2n = 16, 20, and 28.

Keywords

• Steiner triple systems
• Steiner loops
• Sloops

AMS Classification

• 05B07
• 20N05

References

1. M.H. Armanious, Semi-planar Steiner Loops of Cardinality 2n, Discrete Math. 270 (2003) 291-298.
2. M.H. Armanious and E. M. A. Elzayat, Extending Sloops of Cardinality 16 to SQS-Skeins with all Possible Congruence Lattices, Quasigroups and Related Systems, Vol. 12 (2004), 1-12.
3. M.H. Armanious and E. M. A. Elzayat, Subdirectly irreducible Sloops and SQS-Skeins, Quasigroups and Related Systems, Vol. (2007), 1-12.
4. M.H. Armanious, On Steiner loops of cardinality 20, NNTDM, Vol. 12 (2006), 4, 10 – 22.
5. C. Colburn and J. Dinitz, eds., The CRC Handbook of Combinatorial Designs, CRC Press, New York, 1996.
6. J. Doyen, Sur la Sttucture de Certains Systems Triples de Steiner, Math. Z. 111 (1979) 289-300.
7. B. Ganter and H. Werner, Co-ordinatizing Steiner Systems, Ann. Discrete Math. 7 (1980) 3- 24.
8. G. Grätzer, Universal Algebra, Springer – Verlag New York, Heidelberg and Berlin, 2^nd edition, 1997.
9. F. Haray, Graph Theory, Addison-Wesley, Reading, MA (1969).
10. P. Kaski, P.R.J. Östergaärd, S. Topalova, R. Zlatarski, Steiner Triple Systems of order 19 and 21 with Subsystems of Order 7, Disc. Math., Vol. 308, Issue 13, (2008), pp 2732-2741.
11. C.C. Lindner, and A. Rosa, Steiner Quadruple Systems: a Survey Discrete Math.21 (1979) 147-181.
12. R.W. Quackenbush, Varieties of Steiner Loops and Steiner Quasigroups, Can. J. Math. (1976) 1187-1198.