Varieties of solvable pro-p-groups (Q795924)

Let \({\mathfrak B}_ m\) be the variety of groups of exponent m and let \({\mathfrak N}_ c\) be the variety of nilpotent groups with nilpotence degree \(\leq c\). It is proved that the variety of soluble pro-p-groups, not containing all metabelian groups, is a subvariety in \({\mathfrak B}_{p^ n}{\mathfrak N}_ c{\mathfrak B}_{p^ n}\) for some n and c. This theorem is parallel to a result of \textit{M. Kargapolov} and \textit{V. Churkin} [Algebra Logika 10, 651-657 (1971; Zbl 0258.20033)]. The author gives also a description (see Th. 5.1) of the class of finitely generated soluble pro-p-groups of finite rank.