On Multiplication Groups of Middle Bol Loop Related to Left Bol Loop

Abstract

In this paper, we show that an element of Bryant-Schneider group of middle Bol loop $(Q, \circ) $ is an automorphism and pseudo-automorphism. We proved that multiplication groups of middle autotopy nuclei of middle Bol loop $(Q, \circ) $ coincided with the corresponding left Bol loop $(Q, \circ) $ and their multiplication groups were show to be normal subgroups. It was found that the corresponding left Bol loop $(Q, \circ) $ is an indexed two.