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)|
|Number of pages||12|
|Journal||Australasian Journal of Combinatorics|
|State||Published - Aug 20 2015|
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics