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Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite groups are very complex and to understand them requires advanced theory on groups, rings, field and Galois Theory. Know a simple example?

Last, how to prove that that $\mathbb{Z}$ not is a profinite group?

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    Well, there's $\hat{\mathbf Z}$, which is in the same vein.2012-06-20
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    The Galois group of any infinite Galois extension is profinite.2012-06-20
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    Dylan. What is $\mathbb{\widehat{Z}}$?2012-06-20
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    @Andres It's just $\varprojlim \mathbf Z/n\mathbf Z$ directed by divisibility. It follows from the Chinese remainder theorem that $\hat{\mathbf Z} \approx \prod\mathbf Z_p$ [product over all primes], so this may not be so interesting for you.2012-06-20

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