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Describing cyclic extensions of Bol loops. (English) Zbl 1328.20084

It was shown by the author [in S. M. Gagola III, J. Algebra Appl. 13, No. 4, Article ID 1350128 (2014; Zbl 1296.20028)] that if a Moufang loop \(G\) factorizes as \(G=NH\) where \(N\) is a normal subloop and \(H=\langle u\rangle=\langle u^3\rangle\) is a cyclic group then the structure of \(G\) is determined by the binary operation of \(N\), the intersection \(N\cap H\) and how \(u\) permutes the elements of \(N\) as a semi-automorphism of \(N\). Here it is shown that if \(G\) is Moufang with \(H=\langle u\rangle\neq\langle u^3\rangle\) or if \(G\) is a Bol loop, not necessarily Moufang, then the structure of \(G\) is determined by the binary operation of \(N\), the intersection \(N\cap H\), how \(u\) permutes the elements of \(N\) and either of the two binary operations \(x*_1y=(xu)(u\backslash y)\) or \(x*_{-1}y=(xu^{-1})(u^{-1}\backslash y)\) of \(N\). In Section 4, the author presents the general case in Theorem 3 for cyclic extensions.

MSC:

20N05 Loops, quasigroups
20E22 Extensions, wreath products, and other compositions of groups
20E34 General structure theorems for groups

Citations:

Zbl 1296.20028
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