On a method of construction of orthogonal quasigroup systems by means of groups. (Russian. English summary) Zbl 1475.05017
Summary: We suggest a method of construction of orthogonal quasigroups systems and orthogonal Latin squares by means of Frobenius groups. It is proved that this method generalizes many (but not all) existing methods of orthogonal Latin squares construction by means of groups.
Keywords:
quasigroup; Latin square; system of orthogonal quasigroups; orthogonal Latin squares; regular set of permutations; regular automorphism of groupReferences:
| [1] | Belousov V. D., Osnovy teorii kvazigrupp i lup, Nauka, M., 1967 · Zbl 0163.01801 |
| [2] | Belousov V. D., Algebraicheskie seti i kvazigruppy, Shtiintsa, Kishinev, 1971 · Zbl 0245.50005 |
| [3] | Belousov V. D., “Sistemy ortogonalnykh operatsii”, Matem. sb., 77(119):1 (1968), 38-58 · Zbl 0211.33101 |
| [4] | Glukhov M. M., Larin S. V., “Abelevy i metabelevy gruppy s regulyarnymi avtomorfizmami i poluavtomorfizmami”, Matem. zametki, 12:6 (1972), 727-738 · Zbl 0251.20040 |
| [5] | Kargopolov M. I., Merzlyakov Yu. I., Osnovy teorii grupp, Nauka, M., 1972 |
| [6] | Sachkov V. N., Kombinatornye metody diskretnoi matematiki, Nauka, M., 1977 |
| [7] | Kholl M., Kombinatorika, Mir, M., 1970 |
| [8] | Belyavskaya G. B., “Quasigroup power sets and cyclic \(S\)-systems”, Quasigroups and related Systems, 9 (2002), 1-17 · Zbl 1044.20040 |
| [9] | Bose R. C., “On application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares”, Sankhya, 3 (1938), 323-338 |
| [10] | Bose R. C., Nair K. R., “On complete sets of orthogonal Latin squares”, Sankhya, 5 (1941), 361-382 · Zbl 0063.00550 |
| [11] | Bose R. C., Shrikhande S. S., Parker E. T., “Further results on the constructions of mutually orthogonal Latin squares and the falsity Euler‘s conjecture”, Canad. J. Math., 12 (1960), 189-203 · Zbl 0093.31905 · doi:10.4153/CJM-1960-016-5 |
| [12] | Brack R. H., “Finite nets, I”, Canad. J. Math., 3 (1951), 94-107 · Zbl 0042.38802 · doi:10.4153/CJM-1951-012-7 |
| [13] | Brack R. H., A survey of binary systems, Springer-Verlag, Berlin-Heidelberg-Gottingen, 1958 · Zbl 0081.01704 |
| [14] | Denes J., Keedwell A. D., Latin Squares. New Developments in the Theory and Applications, Nord-Holland Publishing Co., Amsterdam, 1981 · Zbl 0715.00010 |
| [15] | Denes J., Mullen G. L., Suchower S. J., “A note on power sets of Latin squares”, J. Combin. Computing, 16 (1994), 27-31 · Zbl 0819.05017 |
| [16] | Denes J., “When is there a Latin power set?”, Amer. Math. Monthly, 104 (1997), 563-565 · Zbl 0884.05022 · doi:10.2307/2975085 |
| [17] | Denes J., Owens P. J., “Some new power sets not based on groups”, J. Combin. Theory Ser. A, 85 (1999), 69-82 · Zbl 0913.05025 · doi:10.1006/jcta.1998.2911 |
| [18] | Denes J., Petroczki P., A digital encrypting communication systems, Hungarian Patent № 201437A, 1990 |
| [19] | Gorenstein D., Finite groups, Harper and Row, New York-Evanston-London, 1968 · Zbl 0185.05701 |
| [20] | Huppert B., Endliche Gruppen, v. I, Springer-Verlag, Berlin-Heidelberg-New York, 1967 · Zbl 0217.07201 |
| [21] | Johnson D. M., Dulmage A. L., Mendelson N. S., “Orthomorphisms of groups and orthogon Latin squares, I”, Canad. J. Math., 13 (1961), 356-372 · Zbl 0097.25102 · doi:10.4153/CJM-1961-031-7 |
| [22] | Keedwell A. D., “On \(R\)-sequenceability and \(R_h\)-sequenceability of groups”, Combinatorics’ 81, Ann. Discrete Math., 78, North-Holland, 1983, 535-548 · Zbl 0502.20008 |
| [23] | Koksma K. K., “Lower bound for the order of a partial transversal in a Latin square”, J. Combinatorial Theory, 7 (1969), 94-95 · Zbl 0172.01504 · doi:10.1016/S0021-9800(69)80009-8 |
| [24] | Laywine C. F., Mullen G. L., Discrete mathematics using Latin squares, John Wiley and Sons, Inc., New York, 1998 · Zbl 0957.05002 |
| [25] | Leakh I. V., On transformations of orthogonal systems of optrations and algebraic nets, Ph. D. thesis, IM AN RM, Kishinev, 1986 |
| [26] | MacNeish H. F., “Euler squares”, Ann. Math., 23 (1921), 221-227 · JFM 49.0041.05 · doi:10.2307/1967920 |
| [27] | Mann H. B., “The construction orthogonal Latin squares”, Ann. Math. Statist., 13 (1942), 418-423 · Zbl 0060.02706 · doi:10.1214/aoms/1177731539 |
| [28] | Norton D. A., “Group of orthogonal row-latin squares”, Pacific J. Math., 2 (1952), 335-341 · Zbl 0047.01703 |
| [29] | Paige L. J., “Complete mappings of finite groups”, Pacific J. Math., 1 (1951), 111-116 · Zbl 0043.02404 |
| [30] | Paige L. J., Hall M., “Complete mappings of finite groups”, Pacific J. Math., 5 (1955), 541-549 · Zbl 0066.27703 |
| [31] | Parker E. T., “Construction of some sets of pairwise orthogonal Latin squares”, Proc. Amer. Math. Soc., 10 (1959), 949-951 · Zbl 0093.02002 · doi:10.1090/S0002-9939-1959-0109789-9 |
| [32] | Parker E. T., “Orthogonal Latin squares”, Proc. Natural Academ. Sci. USA, 45 (1959), 859-962 · Zbl 0086.02201 · doi:10.1073/pnas.45.6.859 |
| [33] | Stevens W., “The Completely orthogonalized Latin squares”, Ann. Eugen., 9 (1939), 269-307 · Zbl 0022.11102 · doi:10.1111/j.1469-1809.1939.tb02214.x |
| [34] | Wanless I. M., “Transversals in Latin squares”, Quasigroups and related systems, 15:1 (2007), 169-190 · Zbl 1158.05015 |
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