M. H. Armanious

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

Volume 17, 2011, Number 2, Pages 52—68

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### 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(2*n*)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(2*n*) to be normal. This result supplies us with the following two facts. First, there is another nonsimple subdirectly irreducible SL(2*n*) 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(2*n*) has a simple sub-SL(*n*) and (*n –* 1)(*n *– 2)/6 sub-SL(8)s passing through a non-unit element, then SL(2*n*) 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(2*n – *1)*s*) in 3 tables for 2*n* = 16, 20, and 28.

### Keywords

- Steiner triple systems
- Steiner loops
- Sloops

### AMS Classification

- 05B07
- 20N05

### References

- M.H. Armanious,
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*Subdirectly irreducible Sloops and SQS-Skeins*, Quasigroups and Related Systems, Vol. (2007), 1-12. - M.H. Armanious,
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## Cite this paper

Armanious, M. H. (2011). On the classes of Steiner loops of small orders. Notes on Number Theory and Discrete Mathematics, 17(2), 52-68.