Isostrophy Bryant-Schneider Group-Invariant of Bol Loops

Tèmítópé Gbóláhàn Jaíyéolá ; Benard Osoba ; Anthony Oyem

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2022), pp. 3-18

Résumé

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.