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diassociative; A-loop; Moufang
In a series of papers from the 1940's and 1950's, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck's colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are Moufang ([5]). In [2] we relaunched this now over 50 year old program by examining conditions under which general --- not necessarily commutative --- diassociative A-loops are, in fact, Moufang. Here, we finish part of the program by characterizing Moufang A-loops. We also investigate simple diassociative A-loops as well as a class of centrally nilpotent diassociative A-loops. These results, {\it in toto}, reveal the distinguished positions two familiar classes of diassociative A-loops --- namely groups and commutative Moufang loops--play in the general theory.
[1] Bruck R.H., Paige L.J.: Loops whose inner mappings are automorphisms. Ann. of Math. 63 (2) (1956), 308-232. MR 0076779 | Zbl 0074.01701
[2] Fuad T.S.R., Phillips J.D., Shen X.R.: On diassociative A-loops. submitted.
[3] Glauberman G.: On loops of odd order II. J. Algebra 8 (1968), 393-414. MR 0222198 | Zbl 0155.03901
[4] Moufang R.: Zur struktur von alternativkorpern. Math. Ann. 110 (1935), 416-430. MR 1512948
[5] Osborn J.M.: A theorem on A-loops. Proc. Amer. Math. Soc. 9 (1959), 347-349. MR 0093555
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