Geodesic
On a~problem from the Kourovka Notebook
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 119-125

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In this article, it is proved that if a group $G$ coincides with its commutator subgroup, is generated by a finite set of classes of conjugate elements, and contains a proper minimal normal subgroup $A$ such that the factor group $G/A$ coincides with the normal closure of one element, then $G$ coincides with the normal closure of an element. From this a positive answer to question 5.52 from the Kourovka Notebook for the group with the condition of minimality on normal subgroups follows. We have found a necessary and sufficient condition for a group coinciding with its commutator subgroup and generated by a finite set of classes of conjugate elements not to coincide with the normal closure of any element. 
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     author = {S. V. Larin},
     title = {On a~problem from the {Kourovka} {Notebook}},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     volume = {11},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a7/}
}
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S. V. Larin. On a~problem from the Kourovka Notebook. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 119-125. http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a7/