M. H. Armanious

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 12, 2006, Number 4, Pages 23–24

**Full paper (PDF, 53 Kb)**

## Details

### Authors and affiliations

M. H. Armanious

*Mathematics Department, Faculty of Scince, Mansoura University,
Mansoura, Egypt *

### Abstract

It is well known that there are five classes of sloops of cardinality 16** ” SL**(16)s” according to the number of sub-**SL**(8)s [4, 6]. In this article, we will show that there are exactly 8 classes of nonsimple sloops and 6 classes of simple sloops of cardinality 20** “SL**(20)s”. Based on the cardinality and the number of (normal) subsloops of** SL**(20), we will construct in section 3 all possible classes of nonsimple** SL**(20)s and in section 4 all possible classes of simple **SL**(20)s. We exhibit the algebraic and combinatoric properties of **SL**(20)s to distinguish each class.

So we may say that there are six classes of **SL**(20)s having one sub-**SL**(10) and n sub-**SL**(8)s for *n* = 0, 1, 2, 3, 4 or 6. All these sloops are subdirectly irreducible having exactly one proper homomorphic image isomorphic to **SL**(2). For *n* = 0, the associated **SL**(20) is a nonsimple subdirectly irreducible having one sub-**SL**(10) and no sub-**SL**(8)s. Indeed, the associated Steiner quasigroup **SQ**(19) of this case supplies us with a new example for a semi-planar **SQ**(19), where the smallest well-known example of semi-planar squags is of cardinality 21 ” cf. [3]”.

It is well known that there is a class of planar Steiner triple systems (**STS**(19)s) due to Doyen [7], where the associated planar **SL**(20) has no sub-**SL**(10) and no sub-**SL**(8). In section 4 we will show that there are other 6 classes of simple **SL**(20)s having n sub-**SL**(8)s for *n* = 0, 1, 2, 3, 4, 6, but no sub-**SL**(10)s. It is well-known that a sub-**SL**(m) of an **SL**(2m) is normal. In the last theorem of this section, we give a necessary and sufficient condition for a sub-**SL**(2) to be normal of an **SL**(2m). Accordingly, we have shown that if a sloop **SL**(20) has a sub-**SL**(10) and 12 sub-**SL**(8), then this sloop is isomorphic to the direct product **SL**(10) × **SL**(2) and if a sloop **SL**(20) has 12 sub-**SL**(8)s and no sub-**SL**(10), then this sloop is a subdirectly irreducible having exactly one proper homomorphic image isomorphic to **SL**(10). In section 5, we describe how can one construct an example for each class of smiple and of nonsimple **SL**(20)s.

### Keywords

- Steiner triple systems
- Steiner loops
- Sloops

### AMS Classification

- Primary: 05B07
- Secondary: 20N05

### References

- M.H. Armanious, On Subdirectly Irreducible Steiner Loops of Cardinality 2
*n*, Beitrage zur Algebra und Geometrie, Vol. 43 (2002), No. 2, 325-331. - M.H. Armanious, Semi-planar Steiner Loops of Cardinality 2
*n*, Discrete Math. 270 (2003) 291-298. - M.H. Armanious, Semi-planar Steiner Quasigroups of Cardinality 3
*n*, Australas. J. Combin., 27 (2003), 13-21. - M. H. Armanious, 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.
- O. Chein, H.O. Pflugfelder and J. D. H. Smith, Quasigroups and Loops, Theory and Applications, Sigma Series in Pure Math., 8, Heldermann Verlag, Berlin, 1990.
- C. Colburn and J. Dinitz, eds., The CRC Handbook of Combinatorial Designs, CRC Press, New York, 1996.
- J. Doyen, Sur la Sttucture de Certains Systems Triples de Steiner, Math. Z. 111 (1979) 289-300.
- B. Ganter, and H. Werner, Co-ordinatizing Steiner Systems, Ann. Discrete Math. 7 (1980) 3- 24.
- G. Grätzer, Universal Algebra, Springer – Verlag New York, Heidelberg and Berlin, 2
^{nd}edition, 1997. - F. Haray, Graph Theory, Addison-Wesley, Reading, MA (1969).
- C. C. Lindner, and A. Rosa, Steiner Quadruple Systems: a Survey Discrete Math.21 (1979) 147-181.
- R.W. Quackenbush, Varieties of Steiner Loops and Steiner Quasigroups, Can. J. Math. (1976) 1187-1198.

## Related papers

## Cite this paper

Armanious, M. H. (2006). On Steiner Loops of cardinality 20. *Notes on Number Theory and Discrete Mathematics*, 12(4), 10-22.