Pro-p group
A profinite group that is a projective limit of finite -groups (cf.
-group). E.g. the additive group of the ring
of
-adic integers is a pro-
-group. In Galois theory pro-
-groups appear as Galois groups of
-extensions of fields.
Let be a pro-
-group. A system of generators of
is a subset
with the properties: 1)
is the smallest closed subgroup of
generated by
; and 2) any neighbourhood of the identity of
contains almost-all (i.e. all except a finite number of) elements of
.
Let be an index set and let
be the abstract free group with system of generators
. The projective limit
of the system of groups
, where
is a normal subgroup of
such that the index of
in
is a power of a number
, while almost-all elements
,
, lie in
, is a pro-
-group, called the free pro-
-group with system of generators
. Every closed subgroup of a free pro-
-group is itself a free pro-
-group. Any pro-
-group
is a quotient group of a free pro-
-group, i.e. there is an exact sequence of homomorphisms of pro-
-groups,
![]() |
where is a suitable free pro-
-group. (This sequence is called a presentation of
by means of
.) A subset
is called a system of relations of
if
is the smallest closed normal subgroup in
containing
and if every open normal subgroup in
contains almost-all elements of
. 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-
-group
have a cohomological interpretation: The first cardinality is the dimension over
of the space
, while the second is the dimension over
of the space
. Here
is regarded as a discrete
-module with a trivial
-action. If
is a finite
-group, then
![]() |
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 ![]() |
[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) |
Pro-p group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pro-p_group&oldid=42149