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Varieties of pro-\(p\)-groups of second order matrices. (English. Russian original) Zbl 0753.20006

Algebra Logic 29, No. 4, 287-301 (1990); translation from Algebra Logika 29, No. 4, 430-451 (1990).
The author proves that an arbitrary proper subvariety of the variety of pro-\(p\)-groups \(V_ 0\) generated by the pro-\(p\)-group of generic matrices of order 2 is contained either in \(N_ cA\) or in \(V_ mN_ cA\), where \(m\) and \(c\geq 1\), \(p\neq 2\). Here \(V_ m\) is the variety of pro-\(p\)-groups generated by the pro-\(p\)-group of generic matrices of order 2 over \(\mathbb{Z}/p^ m\mathbb{Z}\), \(N_ c\) is the variety of all nilpotent pro-\(p\)-groups of class not greater than \(c\), \(A\) is the variety of the all Abelian pro-\(p\)-groups.

MSC:

20E18 Limits, profinite groups
20E10 Quasivarieties and varieties of groups
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[1] A. I. Zubkov, ”The central lower series of a pro-p-group of second-order general matrices,” Preprint No. 731, Akad. Nauk SSSR Sib. Otd., Vychisl. Tsentr., Novosibirsk (1987).
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