Schreier decomposition of loops

Abstract

The aims of this paper are to find algebraic characterizations of Schreier loops and explore the limits of the non-associative generalization of the theory of Schreier extensions. A loop can have Schreier decomposition with respect to a normal subgroup if and only if the subgroup is the middle and right nuclear. In this case the conjugation by elements of the loop induces inner automorphisms on the normal subgroup if and only if the subgroup commutes with a suitable left transversal through the identity. Schreier loops which are Schreier extensions of the same loop by the same normal subgroups are characterized.