On loops of odd order

On nilpotent Moufang loops with central associators

“…The latter has order p 3 for the Moufang loop L of maximal class. By (2),L x,y (z mod L ) = (x, y, z) −1 . Take a generating triple g 1 , g 2 , g 3 of L. Since L is proper, (g 1 , g 2 , g 3 ) = 1.…”

A characterization of the full wreath product

“…For |Ω| = 3, the groups and triples (G, T , π 3 ) satisfying ( * ) have in fact been studied extensively, starting with Glauberman [13] and Doro [6], under the name of groups with triality (or triality groups); see [11,12,16,20], for instance. Such groups need not arise from wreath products, Cartan's triality groups PΩ + 8 (F) : Sym (3), for F a field, furnishing the canonical examples (and the name) of groups with triality.…”

“…Glauberman [13] proved that finite Bruck loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z * -theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime).…”

On 2-Engel groups and Bruck loops

“…In a Moufang loop on which squaring is bijective, Bruck showed that the core is a two-sided quasigroup, isotopic to a Bol loop with the automorphic inverse property, or what is nowadays known as a Bruck loop [3, Theorem VII.5.2]. Glauberman used this Bruck loop structure to obtain many results about finite Moufang loops of odd order, including their solvability and the validity of Sylow's and Hall's Theorems [7,8]. Since then, Bruck loops have gained additional importance through their natural occurrence in many areas, such as geometry, matrix decompositions, and special relativity theory -compare [13], [23,Exercises 2.9,17,18], and [26], for example.…”