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Keywords:
extension of quasigroups; right nucleus; quasigroup with right unit; transversal
extension of quasigroups; right nucleus; quasigroup with right unit; transversal
Summary:
The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma$ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists a normalized left transversal to $G$ in $Q$ contained in the commutant of $G$.
References:
[1] Nagy P.T., Strambach K.: Schreier loops. Czechoslovak Math. J. 58 (133) (2008), 759–786. DOI 10.1007/s10587-008-0050-7 | MR 2455937 | Zbl 1166.20058
[2] Nagy P.T., Stuhl I.: Right nuclei of quasigroup extensions. Comm. Alg. 40 (2012), 1893-1900. DOI 10.1080/00927872.2011.575676
[3] Smith J.D.H., Romanowska A.B.: Post-modern algebra. Wiley, New York, 1999. MR 1673047 | Zbl 0946.00001
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