MATHEMATICS AND MECHANICS
On the Levi Class Generated by the Almost Abelian Quasivariety of Nilpotent Groups
For an arbitrary class of groups M we denote by L(M) the class of all groups G in which the normal closure of every element of G belongs to M. The class L(M) is called the Levi class generated by M. A study of Levi classes should be considered as a step towards studying the structure of groups covered by a system of normal subgroups. Levi classes were introduced under the influence of Levi’s article in which Levi gave the classification of groups with abelian normal closures. R.F. Morse proved that if the class M is a variety of groups then L(M) is also a variety of groups. It follows from the works of A.I. Budkin that if M is a quasivariety, then L(M) is a quasivariety of groups, too. We consider that the quasivariety qH2 is generated by the relatively free group in the class of nilpotent groups of length at most 2 with the exponent commutant of 2. Earlier we found descriptions of the Levi classes generated by the almost abelian quasivarieties of nilpotent groups (i.e. the nonabelian quasivarieties of nilpotent groups in which all proper subquasivarieties are abelian) except L(qH2). In this paper, we continue to explore the Levi classes generated by the almost abelian quasivarieties of nilpotent groups. It is proved that the class L(qH2) contains the nilpotent group of lenght 3.
Key words: group, variety, quasivariety, metabelian group, Levi class
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