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On a construction of all quasifields of order 9. (English) Zbl 0604.51001
S is the set of the 9 2-dimensional vectors as ordered couples of elements of GF(3) whose component-wise addition form the corresponding Cayley table. M is the set of the 48 non-singular \(2\times 2\) matrices over GF(3) which map the 8 non-zero vectors of S into one another. A matrix of M is called dispersing if it does not map any non-zero vector of S into itself. A multiplication operation \(\cdot\) is defined for any 2 non-zero vectors of S, and then GF(9) is derived as \((S,+,\cdot)\) in three ways by generating 3 cyclic groups of matrices of M; a nearfield \((S,+,\cdot)\) by a group of matrices of M isomorphic to the quaternion group;..., and finally a right quasifield \((S,+,\cdot)\) is derived by a set of 8 matrices of M that contains non-dispersing matrices.
In each case the corresponding tables of mappings and multiplication are given by the author, the last multiplication table being isotopic to one of the preceeding 6 such tables.
Reviewer: S.Mandan
MSC:
51A40 Translation planes and spreads in linear incidence geometry
51E15 Finite affine and projective planes (geometric aspects)
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