# Article

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Keywords:
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:
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[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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