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Could someone tell me which is the importance and some applications of the profinite groups?

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  1. A standard application of profinite groups is to deal with Galois Theory for infinite extensions.

    If $K/F$ is an infinite Galois extension, then $\mathrm{Gal}(K/F)$ is the inverse limit of the Galois groups $\mathrm{Gal}(E/F)$, where $F\subseteq E\subseteq K$, $[E:F]$ is finite, and $E$ is Galois over $F$. The Fundamental Theorem now establishes a one-to-one, inclusion reversion correspondence between the closed subgroups of $\mathrm{Gal}(K/F)$ (closed in the profinite topology), and the intermediate fields of $K/F$ that is similar to the one in the finite case.

  2. Another recent application was a proof of the Coclass Conjectures for finite $p$-group. A $p$-group of order $p^n$ and class $c$ is said to have coclass $n-c$. Inspired by Blackburn's classification of $p$-groups of maximal class (which are groups of coclass 1), a project was proposed and carried out to extend to general coclasses. Roughly speaking, the results say that for every $c$ there is a finite set of pro-$p$-groups (the special case of pro-finite groups in which the finite groups in the inverse limit are $p$-groups) such that each $p$-group of coclass $c$ is a perturbation of a quotient of one of these pro-$p$-groups. One proof was done by using properties of analytic pro-$p$-groups

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Arturo in the part 1, which is this Fundamental Theorem ? – Andres Jun 22 '12 at 18:04
@Andres: The Fundamental Theorem of Galois Theory: the inclusion-reversing correspondence between [closed] subgroups of $\mathrm{Gal}(K/F)$ and intermediate fields of $K/F$, which respects the index, identifies normal subextensions with normal subgroups, etc. – Arturo Magidin Jun 22 '12 at 18:19

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