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Hall triple systems and commutative Moufang exponent 3 loops: The case of nilpotence class 2

1984; Elsevier BV; Volume: 36; Issue: 2 Linguagem: Inglês

10.1016/0097-3165(84)90001-3

1096-0899

Robert Roth, D.K. Ray-Chaudhuri,

graph theory and CDMA systems

We study here the structure of the coordinatizing loops of Hall Triple Systems which are Steiner Triple Systems, other than affine geometries, in which every plane is an affine plane. These loops are necessarily non-associative commutative Moufang loops of exponent 3 and we concentrate on those which involve the least complicated computations, namely, those having central nilpotence class 2. After presenting introductory material from Bruck's theory of loops we determine precisely the structure of finitely generated commutative Moufang exponent 3 loops of nilpotence class 2 in terms of a very useful representation. We then give a general method of constructing such loops; and since the triple systems can always be recovered from the loops, this enables us to construct many new Hall Triple Systems. These designs are particularly interesting since they are the only known “non-classical” perfect matroid designs.