Geodesic
All difference family structures arise from groups
Algebra and discrete mathematics, no. 1 (2009), pp. 20-31.

Voir la notice de l'article provenant de la source Math-Net.Ru

Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups.These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of interest to those using nonstandard algebras as bases for defining 2-designs.
Keywords: Algebras, 2-design, difference family
Mots-clés : nonassociative. 
@article{ADM_2009_1_a3,
     author = {Tim Boykett},
     title = {All difference family structures arise from groups},
     journal = {Algebra and discrete mathematics},
     pages = {20--31},
     publisher = {mathdoc},
     number = {1},
     year = {2009},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/ADM_2009_1_a3/}
}
TY  - JOUR
AU  - Tim Boykett
TI  - All difference family structures arise from groups
JO  - Algebra and discrete mathematics
PY  - 2009
SP  - 20
EP  - 31
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/ADM_2009_1_a3/
LA  - en
ID  - ADM_2009_1_a3
ER  - 
%0 Journal Article
%A Tim Boykett
%T All difference family structures arise from groups
%J Algebra and discrete mathematics
%D 2009
%P 20-31
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/ADM_2009_1_a3/
%G en
%F ADM_2009_1_a3
Tim Boykett. All difference family structures arise from groups. Algebra and discrete mathematics, no. 1 (2009), pp. 20-31. https://geodesic-test.mathdoc.fr/item/ADM_2009_1_a3/