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By Luis Ribes

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The most function of those lectures is first to in short survey the basic con­ nection among the illustration thought of the symmetric staff Sn and the speculation of symmetric features and moment to teach how combinatorial tools that come up certainly within the idea of symmetric features bring about effective algorithms to specific numerous prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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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.

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