Varieties of solvable pro-P-groups

Abstract

By a variety of pro~ -groups we will mean a class of pro~ -groups closed under the operations of taking subgroups, continuous homomorphic images, and topological Cartesian products. It has been shown in [2] that a variety of solvable discrete groups not containing all metablian groups is a subvariety in ~ Dt'Z 0<,<~ for some /z and C , where ~ is the variety of groups of period g,, ~6 the variety of nilpotent groups of nilpotency class ~$ The purpose of the present article is to prove the analog of this result for varieties of solvable pro-p-groups. In Sec. 3 we will prove THEOREM A~ A variety of solvable pro-p-groups not containing all metabelian groups is a subvariety of the variety ~ c ~ p ~ for some oc and C. In Sec. 4, Theorem B is proved generalizing Theorem A to the case of subvarieties of a product of locally finite or abelian varieties. In Sec. 5 we give a description of finitely generated pro-p-groups of finite rank. i. Differentiations of Completed Group Algebras