By Muratov M.A.

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**Example text**

Then we denote the corresponding free pro-C group F (X ∪. {∗}, ∗) by F (X), which satisfies an obvious universal property as above, but where the maps are not anymore maps of pointed spaces. These more general free pro-C groups are often very useful when trying to describe the subgroup structure of (normal) subgroups of a free pro-C group (see Chapters 3 and 8 in [5]) . For example, if F = F (x, y) is the free profinite group of rank 2, then the closed normal subgroup of F generated by x can be easily ˆ described as a free profinite on a space homeomorphic to Z.

Note that ϕ is the restriction of ϕ1 to G. Since F is C-projective, there exists a continuous homomorphism ϕ¯1 : F −→ A such that αϕ¯1 = ϕ1 . 3), as needed. , extension-closed varieties, the distinction between ‘projective’ and ‘C-projective’ is non-existent. 7). 18. Let C be a saturated variety of finite groups and let G be a pro-C group. Then the following conditions on G are equivalent: (a) G is a C-projective group; (b) G is a projective group; (c) cd(G) ≤ 1. 19 (cf. 4). Let G be a pro-p group.

Since X is a set converging to 1 and ϕ and σ are continuous, the mapping ϕ1 converges to 1. Therefore, ϕ1 extends to a continuous homomorphism ϕ¯ : G −→ A with αϕ¯ = ϕ. Finally note that ϕ¯ is onto since ϕ1 (X) generates A. 7 in [5]. 14, we get the following characterization of free pro-C groups of infinite countable rank. 17 (Iwasawa). Let C be a variety of finite groups and let G be a proC group with w0 (G) = ℵ0 . Then G is a free pro-C group on a countably infinite set converging to 1 if and only if every embedding problem of pro-C groups of the form G GK 1 has a solution whenever A is finite.