Profinite group

 ( Infinite Group Theory ) 

There is a natural homomorphism $\backslash eta\; :\; G\; \backslash to\; \backslash widehat\{G\}$, and the image of $G$ under this homomorphism is dense set in $\backslash widehat\{G\}$. The homomorphism $\backslash eta$ is injective if and only if the group $G$ is residually finite (i.e., $\backslash bigcap\; N\; =\; 1$, where the intersection runs through all normal subgroups $N$ of finite index).

The homomorphism $\backslash eta$ is characterized by the following universal property: given any profinite group $H$ and any continuous group homomorphism $f\; :\; G\; \backslash rightarrow\; H$ where $G$ is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous group homomorphism $g\; :\; \backslash widehat\{G\}\; \backslash rightarrow\; H$ with $f\; =\; g\; \backslash eta$.

Conversely, any group $G$ satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit $\backslash varprojlim\; G/N$ where $N$ ranges through the open of $G$ ordered by (reverse) inclusion. If $G$ is topologically finitely generated then it is in addition equal to its own profinite completion.

• Finite groups are profinite, if given the discrete topology.
• The group of p-adic number $\backslash Z\_p$ under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups $\backslash Z/p^n\backslash Z$ where $n$ ranges over all and the natural maps $\backslash Z/p^n\backslash Z\; \backslash to\; \backslash Z/p^m\backslash Z$ for $n\; \backslash ge\; m.$ The topology on this profinite group is the same as the topology arising from the $p$-adic valuation on $\backslash Z\_p.$
• The group of profinite integers $\backslash widehat\{\backslash Z\}$ is the profinite completion of $\backslash Z.$ In detail, it is the inverse limit of the finite groups $\backslash Z/n\backslash Z$ where $n\; =\; 1,2,3,\backslash dots$ with the modulo maps $\backslash Z/n\backslash Z\; \backslash to\; \backslash Z/m\backslash Z$ for $m\backslash ,|\backslash ,n.$ This group is the product of all the groups $\backslash Z\_p,$ and it is the absolute Galois group of any finite field.
• The Galois theory of of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if $L\; /\; K$ is a Galois extension, consider the group $G\; =\; \backslash operatorname\{Gal\}(L\; /\; K)$ consisting of all field automorphisms of $L$ that keep all elements of $K$ fixed. This group is the inverse limit of the finite groups $\backslash operatorname\{Gal\}(F\; /\; K),$ where $F$ ranges over all intermediate fields such that $F\; /\; K$ is a Galois extension. For the limit process, the restriction homomorphisms $\backslash operatorname\{Gal\}(F\_1\; /\; K)\; \backslash to\; \backslash operatorname\{Gal\}(F\_2\; /\; K)$ are used, where $F\_2\; \backslash subseteq\; F\_1.$ The topology obtained on $\backslash operatorname\{Gal\}(L\; /\; K)$ is known as the Krull topology after Wolfgang Krull. showed that profinite group is isomorphic to one arising from the Galois theory of field $K,$ but one cannot (yet) control which field $K$ will be in this case. In fact, for many fields $K$ one does not know in general precisely which occur as Galois groups over $K.$ This is the inverse Galois problem for a field $K.$ (For some fields $K$ the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.) Not every profinite group occurs as an absolute Galois group of a field.Fried & Jarden (2008) p. 497
• The étale fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in general not profinite: for any prescribed group, there is a 2-dimensional CW complex whose fundamental group equals it.
• The automorphism group of a locally finite rooted tree is profinite.

• Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is Exact functor on the category of profinite groups. Further, being profinite is an extension property.
• Every closed set subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If $N$ is a closed normal subgroup of a profinite group $G,$ then the factor group $G\; /\; N$ is profinite; the topology arising from the profiniteness agrees with the quotient topology.
• Since every profinite group $G$ is compact Hausdorff, there exists a Haar measure on $G,$ which allows us to measure the "size" of subsets of $G,$ compute certain probabilities, and integral functions on $G.$
• A subgroup of a profinite group is open if and only if it is closed and has finite index.
• According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely generated profinite group (that is, a profinite group that has a dense set finitely generated subgroup) the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely generated pro-p group. The proof uses the classification of finite simple groups.
• As an easy corollary of the Nikolov–Segal result above, surjective discrete group homomorphism $\backslash varphi\; :\; G\; \backslash to\; H$ between profinite groups $G$ and $H$ is continuous as long as $G$ is topologically finitely generated. Indeed, any open subgroup of $H$ is of finite index, so its preimage in $G$ is also of finite index, and hence it must be open.
• Suppose $G$ and $H$ are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism $\backslash iota.$ Then $\backslash iota$ is bijective and continuous by the above result. Furthermore, $\backslash iota^\{-1\}$ is also continuous, so $\backslash iota$ is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its structure.

By applying Pontryagin duality, one can see that abelian group profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian .

Projectivity for a profinite group $G$ is equivalent to either of the two properties:

• the cohomological dimension $\backslash operatorname\{cd\}(G)\; \backslash leq\; 1;$
• for every prime $p$ the Sylow $p$-subgroups of $G$ are free pro-$p$-groups.

Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries.Fried & Jarden (2008) pp. 208,545

A topological group $G$ is procyclic if and only if $G\; \backslash cong\; \{\backslash textstyle\backslash prod\backslash limits\_\{p\backslash in\; S\}\}\; G\_p$ where $p$ ranges over some set of $S$ and $G\_p$ is isomorphic to either $\backslash Z\_p$ or $\backslash Z/p^n\; \backslash Z,\; n\; \backslash in\; \backslash N.$

• (2025). 9783540772699, Springer-Verlag. ISBN 9783540772699
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• . Review of several books about profinite groups.
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