On 2-Engel groups and Bruck loops. (Q2856531)

In a Moufang loop \(L\) on which the squaring is bijective Bruck showed that the core operation \(a\circ b=(b a^{-1})b\) is isotopic to a Bol loop with the automorphic inverse property, i.e. to a Bruck loop (Theorem VII.5.2 in [\textit{R. H. Bruck}, A survey of binary systems. Berlin-Göttingen-Heidelberg: Springer-Verlag (1958; Zbl 0081.01704)]). In this paper the author extends the idea of Bruck's construction to a quadratically closed Bol loop or group (where the squaring operation is surjective) satisfying the \(2\)-Engel condition. Furthermore, a process, called bijectivization, is applied to obtain a bijective version of the squaring map on the so-called history space, a space of doubly-infinite sequences of loop elements such that each element in the sequence is succeeded by its square. If the original loop is Moufang or (right) Bol, then the history space forms a Bruck loop.