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A [[Profinite group|profinite group]] that is a projective limit of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748802.png" />-groups (cf. [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748803.png" />-group]]). E.g. the additive group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748805.png" />-adic integers is a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748806.png" />-group. In Galois theory pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748807.png" />-groups appear as Galois groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748808.png" />-extensions of fields.
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A [[Profinite group|profinite group]] that is a [[projective limit]] of finite [[P-group|$p$-group]]s. E.g. the additive group of the ring $\mathbf{Z}_p$ of $p$-adic integers is a pro-$p$-group. In Galois theory pro-$p$-groups appear as [[Galois group]]s of $p$-extensions of fields.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p0748809.png" /> be a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488010.png" />-group. A system of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488012.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488013.png" /> with the properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488014.png" /> is the smallest closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488015.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488016.png" />; and 2) any neighbourhood of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488017.png" /> contains almost-all (i.e. all except a finite number of) elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488018.png" />.
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Let $G$ be a pro-$p$-group. A system of generators of $G$ is a subset $E \subset G$ with the properties: 1) $G$ is the smallest closed subgroup of $G$ generated by $E$; and 2) any neighbourhood of the identity of $G$ contains almost-all (i.e. all except a finite number of) elements of $E$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488019.png" /> be an index set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488020.png" /> be the abstract free group with system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488021.png" />. The projective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488022.png" /> of the system of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488024.png" /> is a [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488025.png" /> such that the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488027.png" /> is a power of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488028.png" />, while almost-all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488030.png" />, lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488031.png" />, is a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488032.png" />-group, called the free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488034.png" />-group with system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488035.png" />. Every closed subgroup of a free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488036.png" />-group is itself a free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488037.png" />-group. Any pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488038.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488039.png" /> is a quotient group of a free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488040.png" />-group, i.e. there is an exact sequence of homomorphisms of pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488041.png" />-groups,
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Let $I$ be an index set and let $F_I$ be the abstract free group with system of generators $\{a_i : i \in I\}$. The projective limit $F(I)$ of the system of groups $F_I/N$, where $N$ is a [[normal subgroup]] of $F_I$ such that the index of $N$ in $F_I$ is a power of a prime number $p$, while almost-all elements $a_i$, $i\in I$, lie in $N$, is a pro-$p$-group, called the free pro-$p$-group with system of generators $\{a_i\}$. Every closed subgroup of a free pro-$p$-group is itself a free pro-$p$-group. Any pro-$p$-group $G$ is a quotient group of a free pro-$p$-group, i.e. there is an exact sequence of homomorphisms of pro-$p$-groups,
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$$
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1 \rightarrow R \rightarrow F \stackrel{\alpha}{\rightarrow} G \rightarrow 1
 +
$$
 +
where $F$ is a suitable free pro-$p$-group. (This sequence is called a presentation of $G$ by means of $F$.) A subset $E\subset R$ is called a system of relations of $G$ if $R$ is the smallest closed normal subgroup in $F$ containing $E$ and if every open normal subgroup in $R$ contains almost-all elements of $E$. The cardinalities of a minimal (with respect to inclusion) set of generators and a minimal system of relations of a corresponding presentation of a pro-$p$-group $G$ have a cohomological interpretation: The first cardinality is the dimension over $\mathbf{F}_p$ of the space $H^1(G) = H^1(G,\mathbf{Z}/p\mathbf{Z})$, while the second is the dimension over $\mathbf{F}_p$ of the space $H^2(G) = H^2(G,\mathbf{Z}/p\mathbf{Z})$. Here $\mathbf{Z}/p\mathbf{Z}$ is regarded as a discrete $G$-module with a trivial $G$-action. If $G$ is a finite $p$-group, then
 +
$$
 +
4 \dim H^2(G) \ge (\dim H^1(G) - 1)^2 \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488042.png" /></td> </tr></table>
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This result implies a negative solution to the classical class field tower problem (cf. [[Tower of fields]]) [[#References|[4]]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488043.png" /> is a suitable free pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488044.png" />-group. (This sequence is called a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488047.png" /> by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488048.png" />.) A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488049.png" /> is called a system of relations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488051.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488052.png" /> is the smallest closed normal subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488053.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488054.png" /> and if every open normal subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488055.png" /> contains almost-all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488056.png" />. The cardinalities of a minimal (with respect to inclusion) set of generators and a minimal system of relations of a corresponding presentation of a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488057.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488058.png" /> have a cohomological interpretation: The first cardinality is the dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488059.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488060.png" />, while the second is the dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488061.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488062.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488063.png" /> is regarded as a discrete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488064.png" />-module with a trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488065.png" />-action. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488066.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488067.png" />-group, then
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====References====
 
+
<table>
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488068.png" /></td> </tr></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Cohomologie Galoisienne" , Springer  (1964)</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  H. Koch,  "Galoissche Theorie der $p$>-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)</TD></TR>
This result implies a negative solution to the classical class field tower problem (cf. [[Tower of fields|Tower of fields]]) [[#References|[4]]].
+
<TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top"> E.S. Golod,  I.R. Shafarevich,  "On the class field tower" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' :  2  (1964)  pp. 261–272  (In Russian)</TD></TR>
 +
</table>
  
====References====
+
{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Cohomologie Galoisienne" , Springer  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Koch,  "Galoissche Theorie der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074880/p07488069.png" />-Erweiterungen" , Deutsch. Verlag Wissenschaft.  (1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.S. Golod,  I.R. Shafarevich,  "On the class field tower"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' :  2  (1964)  pp. 261–272  (In Russian)</TD></TR></table>
 

Latest revision as of 19:45, 21 October 2017

A profinite group that is a projective limit of finite $p$-groups. E.g. the additive group of the ring $\mathbf{Z}_p$ of $p$-adic integers is a pro-$p$-group. In Galois theory pro-$p$-groups appear as Galois groups of $p$-extensions of fields.

Let $G$ be a pro-$p$-group. A system of generators of $G$ is a subset $E \subset G$ with the properties: 1) $G$ is the smallest closed subgroup of $G$ generated by $E$; and 2) any neighbourhood of the identity of $G$ contains almost-all (i.e. all except a finite number of) elements of $E$.

Let $I$ be an index set and let $F_I$ be the abstract free group with system of generators $\{a_i : i \in I\}$. The projective limit $F(I)$ of the system of groups $F_I/N$, where $N$ is a normal subgroup of $F_I$ such that the index of $N$ in $F_I$ is a power of a prime number $p$, while almost-all elements $a_i$, $i\in I$, lie in $N$, is a pro-$p$-group, called the free pro-$p$-group with system of generators $\{a_i\}$. Every closed subgroup of a free pro-$p$-group is itself a free pro-$p$-group. Any pro-$p$-group $G$ is a quotient group of a free pro-$p$-group, i.e. there is an exact sequence of homomorphisms of pro-$p$-groups, $$ 1 \rightarrow R \rightarrow F \stackrel{\alpha}{\rightarrow} G \rightarrow 1 $$ where $F$ is a suitable free pro-$p$-group. (This sequence is called a presentation of $G$ by means of $F$.) A subset $E\subset R$ is called a system of relations of $G$ if $R$ is the smallest closed normal subgroup in $F$ containing $E$ and if every open normal subgroup in $R$ contains almost-all elements of $E$. The cardinalities of a minimal (with respect to inclusion) set of generators and a minimal system of relations of a corresponding presentation of a pro-$p$-group $G$ have a cohomological interpretation: The first cardinality is the dimension over $\mathbf{F}_p$ of the space $H^1(G) = H^1(G,\mathbf{Z}/p\mathbf{Z})$, while the second is the dimension over $\mathbf{F}_p$ of the space $H^2(G) = H^2(G,\mathbf{Z}/p\mathbf{Z})$. Here $\mathbf{Z}/p\mathbf{Z}$ is regarded as a discrete $G$-module with a trivial $G$-action. If $G$ is a finite $p$-group, then $$ 4 \dim H^2(G) \ge (\dim H^1(G) - 1)^2 \ . $$

This result implies a negative solution to the classical class field tower problem (cf. Tower of fields) [4].

References

[1] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964)
[2] H. Koch, "Galoissche Theorie der $p$>-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)
[3] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[4] E.S. Golod, I.R. Shafarevich, "On the class field tower" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 261–272 (In Russian)
How to Cite This Entry:
Pro-p group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pro-p_group&oldid=15018
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article