Previous |  Up |  Next

Article

Keywords:
Moufang loops; RA loops; alternative rings; minimal nonassociativity
Summary:
We investigate finite Moufang loops with a unique nonidentity commutator which are not associative, but all of whose proper subloops are associative. Curiously, perhaps, such loops turn out to be ``ring alternative'', in the sense that their loop rings are alternative rings.
References:
[Bru58] Bruck R.H.: A survey of binary systems. Ergeb. Math. Grenzgeb., vol. 20, Springer-Verlag, 1958. MR 0093552 | Zbl 0141.01401
[CG86] Chein O., Goodaire E.G.: Loops whose loop rings are alternative. Comm. Algebra 14 (1986), 2 293-310. MR 0817047 | Zbl 0582.17015
[CG90a] Chein O., Goodaire E.G.: Loops whose loop rings in characteristic $2$ are alternative. Comm. Algebra 18 (1990), 3 659-688. MR 1052760 | Zbl 0718.20034
[CG90b] Chein O., Goodaire E.G.: Moufang loops with a unique nonidentity commutator (associator, square). J. Algebra 130 (1990), 2 369-384. MR 1051308 | Zbl 0695.20040
[Che74] Chein O.: Moufang loops of small order I. Trans. Amer. Math. Soc. 188 (1974), 31-51. MR 0330336 | Zbl 0286.20088
[Che78] Chein O.: Moufang loops of small order. Mem. Amer. Math. Soc. 13 (1978), 197 1-131. MR 0466391 | Zbl 0378.20053
[CR72] Chein O., Robinson D.A.: An ``extra'' law for characterizing Moufang loops. Proc. Amer. Math. Soc. 33 (1972), 29-32. MR 0292987 | Zbl 0215.40302
[Fen68] Fenyves F.: Extra loops I. Publ. Math. Debrecen 15 (1968), 235-238. MR 0237695 | Zbl 0172.02401
[GJM96] Goodaire E.G., Jespers E., Polcino Milies C.: Alternative loop rings. North-Holland Math. Studies, vol. 184, Elsevier, Amsterdam, 1996. MR 1433590 | Zbl 0878.17029
[GP87] Goodaire E.G., Parmenter M.M.: Semi-simplicity of alternative loop rings. Acta Math. Hungar. 50 (1987), 3-4 241-247. MR 0918159 | Zbl 0634.17014
[JLM95] Jespers E., Leal G., Polcino Milies C.: Classifying indecomposable RA loops. J. Algebra 176 (1995), 5057-5076. MR 1351625
[MM03] Miller G.A., Moreno H.C.: Nonabelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), 398-404. MR 1500650
[Pfl90] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Heldermann Verlag, Berlin, 1990. MR 1125767 | Zbl 0715.20043
Partner of
EuDML logo