FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 3, PAGES 119-125

On a problem from the Kourovka Notebook

S. V. Larin

Abstract

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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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Last modified: September 14, 2005