ON SCHREIER EXTENSION OF LOOPS

Antoni Chronowski
1982 Demonstratio Mathematica  
ON SCHREIER EXTENSION OF LOOPS Schreier presented in the paper [2] 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 [1]. Let
more » ... e paper [1]. 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.
doi:10.1515/dema-1982-0109 fatcat:kb3nsdpq2bhy7kneed6wbeinmq