On loop extensions satisfying one single identity and cohomology of loops

Abstract

ABSTRACTIn this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3] and of Kenneth and Leedham-Green [2]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.