Bryant-Schneider Group of Basarab Loop

A loop (Q,∘) is called Basarab loop if it is both a left and a right Basarab loop; (x∘yxρ) ∘xz = x∘yz and yx∘ (xλz ∘x) = yz ∘x hold for all x,y,z ∈ Q respectively. In this paper, the characterizations of the Bryant-Schneider group of a Basarab loop are studied using the left and right Basarab loop identities. It is shown that the element, xλ (xρ) is in the left (right) nucleus if and only if the middle inner map Tx (inverse Tx−1) is an automorphism. It is revealed that every crypto-automorphism of a Basarab loop is an element of the Bryant-Schneider group. Some related algebraic properties were also characterized. Furthermore, elements of the Bryant-Schneider group of a Basarab loop in terms of pseudo-automorphism and automorphism are also characterized. A subgroup of the Bryant-Schneider group, characterized by the Basarab loop, is established. Finally, a right pseudo-automorphic characterization of the isotopy-isomorphy of a Basarab loop is carried out.