Cycles in Algebraic Systems

descriptionPublicationkeyboard_double_arrow_right Article , Other literature type 01 Dec 1956Publisher:JSTORJournal:Proceedings of the American Mathematical Society, volume 7, page 999 (issn: 0002-9939,

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Authors: Norton, D. A.; Stein, Sherman K.;

doi: 10.2307/2033027 , 10.1090/s0002-9939-1956-0082989-x

Cycles in Algebraic Systems

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Abstract

2. Cycles and their terminology. Let L be an n Xn Latin square or, alternatively, let Q be a quasigroup of order n. Let M be the set of n2 ordered triplets ijk, where k is the entry in the ith row and jth column of L. If S is the set of n2 ordered pairs ij, and if rir: M->S is the projection parallel to the ith coordinate (for example, ir2(ijk) =ik), then for each i, 1_i_3, 7ri is onto S (or equivalently is one-one). There is clearly a one-one correspondence between Latin squares L of order n and sets M of n2 ordered triplets for which the iri are all onto S. Let T: S-S be the involution defined by T(j) = (ji) and Pi: M-M be 7r 'Thrs. For each i, 1

Keywords

group theory

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