On Levi quasivarieties generated by nilpotent groups. (English. Russian original) Zbl 0956.20015
Sib. Math. J. 41, No. 2, 218-223 (2000); translation from Sib. Mat. Zh. 41, No. 2, 270-277 (2000).
Let \(\mathcal M\) be a class of groups. By \(L({\mathcal M})\) we denote the class of groups \(G\) such that, for every \(x\in G\), the normal closure \((x)^G\) belongs to \(\mathcal M\). This yields the notion of the Levi class generated by a class of groups. By \(q{\mathcal M}\) the authors denote the quasivariety generated by \(\mathcal M\). The authors prove the following theorem: Let \(\mathcal K\) be an arbitrary set of nilpotent groups of class \(2\) without elements of order \(2\). Suppose that, in every group in \(\mathcal K\), the centralizer of each element outside the center of the group is an Abelian group. Then, for \({\mathcal M}=q{\mathcal K}\), we have \(L({\mathcal M})\subseteq{\mathcal N}_3\).
Reviewer: K.N.Ponomarev (Novosibirsk)
MSC:
| 20E10 | Quasivarieties and varieties of groups |
| 20F18 | Nilpotent groups |
| 20F45 | Engel conditions |
| 08C15 | Quasivarieties |
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