M. H. Armanious

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

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

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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

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## Cite this paper

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