### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 170-181 |

Number of pages | 12 |

Journal | Australasian Journal of Combinatorics |

Volume | 63 |

Issue number | 1 |

State | Published - Aug 20 2015 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*63*(1), 170-181.

**Steiner loops satisfying moufang’s theorem.** / Colbourn, Charles; Giuliani, Maria De Lourdes Merlini; Rosa, Alexander; Stuhl, Izabella.

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 63, no. 1, pp. 170-181.

}

TY - JOUR

T1 - Steiner loops satisfying moufang’s theorem

AU - Colbourn, Charles

AU - Giuliani, Maria De Lourdes Merlini

AU - Rosa, Alexander

AU - Stuhl, Izabella

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

VL - 63

SP - 170

EP - 181

JO - Australasian Journal of Combinatorics

JF - Australasian Journal of Combinatorics

SN - 1034-4942

IS - 1

ER -