Geodesic
Quasigroups arisen by right nuclear extension
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 391-395.

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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 $G$.
Classification : 20N05
Keywords: extension of quasigroups; right nucleus; quasigroup with right unit; transversal 
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     title = {Quasigroups arisen by right nuclear extension},
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Nagy, Péter T.; Stuhl, Izabella. Quasigroups arisen by right nuclear extension. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 391-395. http://geodesic.mathdoc.fr/item/CMUC_2012__53_3_a5/