Isostrophy Bryant-Schneider Group-Invariant of Bol Loops

Abstract

In the recent past, Grecu and Syrbu (in no order of preference) have jointly and individually reported some results on isostrophy invariants of Bol loops. Also, the Bryant-Schneider group of a loop has been found important in the study of the isotopy-isomorphy of some varieties of loops (e.g. Bol loops, Moufang loops, Osborn loops). In this current work, the Bryant-Schneider group of a middle Bol loop was linked with some of the isostrophy-group invariance results of Grecu and Syrbu. In particular, it was shown that some subgroups of the Bryant-Schneider group of a middle Bol loop are equal (or isomorphic) to the automorphism and pseudo-aumorphism groups of its corresponding right (left) Bol loop. Some elements of the Bryant-Schneider group of a middle Bol loop were shown to induce automorphisms and middle pseudo-automorphisms. It was discovered that if a middle Bol loop is of exponent 2, then, its corresponding right (left) Bol loop is a left (right) G-loop.