Some Constructions of Non-Commutative Latin Squares of Order n
Let n be an integer. We show the existence of at least
three non-isomorphic non-commutative Latin squares of order n
which are embeddable in groups when n ≥ 5 is odd. By using a
similar construction for the case when n ≥ 4 is even, we show that
certain non-commutative Latin squares of order n are not embeddable
in groups.
[1] H. V. Chen, A. Y. M. Chin, and S. Sharmini, Constructions of noncommutative
generalized latin squares of order 5. Proceedings of the
6th IMT-GT Conference on Mathematics, Statistics and its Applications
(ICMSA2010) (Kuala Lumpur, Malaysia), November 2010, pp. 120-130.
[2] H. V. Chen, A. Y. M. Chin, and S. Sharmini. On non-commuting subsets
of certain types in groups. Research Report No. 6/2010, Institute of
Mathematical Sciences, Faculty of Science, University of Malaya.
[3] G. A. Freiman. On two- and three-element subsets of groups. Aequationes
Mathematicae, 22:140-152, 1981.
[4] G. A. Freiman. Foundations of a structural theory of set addition,
Translation from Russian. Translations of Math, Monographs, Vol. 37,
Providence, R.I.: Amer. Math. Soc. VII, 1973.
[5] J. J. H. Tan. Some properties of subsets of finite groups. MSc Thesis,
Institute of Mathematical Sciences, University of Malaya, 2004.
[1] H. V. Chen, A. Y. M. Chin, and S. Sharmini, Constructions of noncommutative
generalized latin squares of order 5. Proceedings of the
6th IMT-GT Conference on Mathematics, Statistics and its Applications
(ICMSA2010) (Kuala Lumpur, Malaysia), November 2010, pp. 120-130.
[2] H. V. Chen, A. Y. M. Chin, and S. Sharmini. On non-commuting subsets
of certain types in groups. Research Report No. 6/2010, Institute of
Mathematical Sciences, Faculty of Science, University of Malaya.
[3] G. A. Freiman. On two- and three-element subsets of groups. Aequationes
Mathematicae, 22:140-152, 1981.
[4] G. A. Freiman. Foundations of a structural theory of set addition,
Translation from Russian. Translations of Math, Monographs, Vol. 37,
Providence, R.I.: Amer. Math. Soc. VII, 1973.
[5] J. J. H. Tan. Some properties of subsets of finite groups. MSc Thesis,
Institute of Mathematical Sciences, University of Malaya, 2004.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63660", author = "H. V. Chen and A. Y. M. Chin and S. Sharmini", title = "Some Constructions of Non-Commutative Latin Squares of Order n", abstract = "Let n be an integer. We show the existence of at least
three non-isomorphic non-commutative Latin squares of order n
which are embeddable in groups when n ≥ 5 is odd. By using a
similar construction for the case when n ≥ 4 is even, we show that
certain non-commutative Latin squares of order n are not embeddable
in groups.", keywords = "group, Latin square, embedding.", volume = "5", number = "7", pages = "1084-3", }