ON SCHREIER EXTENSION OF LOOPS

Antoni Chronowski
1982 Demonstratio Mathematica
ON SCHREIER EXTENSION OF LOOPS Schreier presented in the paper  the solution of the problem of determining all extensions of an arbitrary group G by means of an arbitrary group H. In the present paper we will develope a generalization of Schreier's theory of extensions of groups, which will allows us to determine all extensions of any loop X by means of a loop H. Definitions of such notions as loop, subloop, coset, normal subloop, quotient loop will be used according to the paper . Let
more » ... e paper . Let L/H be a quotient loop of a loop L modulo H. A function s: L/H L is called a selector, whenever it satisfies the following condition /\ s(M) 6 M. MeL/H As for groups we can formulate the definition of an extension of a loop. Definition. mi extension of a loop K by means of a loopj L is defined as a loop 2 , which satisfies the following conditions: (i) K is a normal subloop of a loop 2, (ii) a quotient loop 2 /K and a loop L are isomorphic.