Pro-p group

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)