TY - JOUR
T1 - Steiner loops satisfying moufang’s theorem
AU - Colbourn, Charles J.
AU - Giuliani, Maria De Lourdes Merlini
AU - Rosa, Alexander
AU - Stuhl, Izabella
N1 - Publisher Copyright:
© 2015,University of Queensland. All rights reserved.
PY - 2015/8/20
Y1 - 2015/8/20
N2 - A loop satisfies Moufang’s theorem whenever the subloop generated by three associating elements is a group. Moufang loops (loops that satisfy the Moufang identities) satisfy Moufang’s theorem, but it is possible for a loop that is not Moufang to nevertheless satisfy Moufang’s theorem. Steiner loops that are not Moufang loops are known to arise from Steiner triple systems in which some triangle does not generate a subsystem of order 7, while Steiner loops that do not satisfy Moufang’s theorem are shown to arise from Steiner triple systems in which some quadrilateral (Pasch configuration) does not generate a subsystem of order 7. Consequently, the spectra of values of v for which a Steiner loop exists are determined when the loop is also Moufang; when the loop is not Moufang yet satisfies Moufang’s theorem; and when the loop does not satisfy Moufang’s theorem. Furthermore, examples are given of non-commutative loops that satisfy Moufang’s theorem yet are not Moufang loops.
AB - A loop satisfies Moufang’s theorem whenever the subloop generated by three associating elements is a group. Moufang loops (loops that satisfy the Moufang identities) satisfy Moufang’s theorem, but it is possible for a loop that is not Moufang to nevertheless satisfy Moufang’s theorem. Steiner loops that are not Moufang loops are known to arise from Steiner triple systems in which some triangle does not generate a subsystem of order 7, while Steiner loops that do not satisfy Moufang’s theorem are shown to arise from Steiner triple systems in which some quadrilateral (Pasch configuration) does not generate a subsystem of order 7. Consequently, the spectra of values of v for which a Steiner loop exists are determined when the loop is also Moufang; when the loop is not Moufang yet satisfies Moufang’s theorem; and when the loop does not satisfy Moufang’s theorem. Furthermore, examples are given of non-commutative loops that satisfy Moufang’s theorem yet are not Moufang loops.
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M3 - Article
AN - SCOPUS:84939506481
SN - 1034-4942
VL - 63
SP - 170
EP - 181
JO - Australasian Journal of Combinatorics
JF - Australasian Journal of Combinatorics
IS - 1
ER -