Geodesic
On loops that are abelian groups over the nucleus and Buchsteiner loops
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 197-208
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We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z(Q)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $\operatorname{Inn} Q$ are abelian groups, then $Q/Z(Q)$ is a CC loop, consequently $Q$ has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z(Q)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $\operatorname{Inn} Q$ are abelian groups, then $Q/Z(Q)$ is a CC loop, consequently $Q$ has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
Classification : 20D10, 20D15, 20N05
Keywords: conjugacy closed loops; Buchsteiner loops 
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     author = {Cs\"org\"o, Piroska},
     title = {On loops that are abelian groups over the nucleus and {Buchsteiner} loops},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {197--208},
     year = {2008},
     volume = {49},
     number = {2},
     mrnumber = {2426885},
     zbl = {1192.20052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2008_49_2_a2/}
}
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Csörgö, Piroska. On loops that are abelian groups over the nucleus and Buchsteiner loops. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 197-208. http://geodesic.mathdoc.fr/item/CMUC_2008_49_2_a2/