Pro-p group explained
such that for any
open normal subgroup
the
quotient group
is a
p-group. Note that, as profinite groups are
compact, the open subgroups are exactly the
closed subgroups of finite
index, so that the
discrete quotient group is always finite.
Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups.
The best-understood (and historically most important) class of pro-p groups is the p-adic analytic groups: groups with the structure of an analytic manifold over
such that group multiplication and inversion are both analytic functions.The work of
Lubotzky and Mann, combined with
Michel Lazard's solution to
Hilbert's fifth problem over the
p-adic numbers, shows that a pro-
p group is
p-adic analytic if and only if it has finite
rank, i.e. there exists a positive integer
such that any closed subgroup has a topological generating set with no more than
elements. More generally it was shown that a finitely generated profinite group is a compact p-adic
Lie group if and only if it has an open subgroup that is a uniformly powerful pro-p-group.
The Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number p and any positive integer r, there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of directed coclass graphs.
Examples
Zp=\displaystyle\varprojlimZ/pnZ.
of invertible
n by
n matrices over
has an open subgroup
U consisting of all matrices congruent to the
identity matrix modulo
. This
U is a pro-
p group. In fact the
p-adic analytic groups mentioned above can all be found as closed subgroups of
for some integer
n,
- Any finite p-group is also a pro-p-group (with respect to the constant inverse system).
- Fact: A finite homomorphic image of a pro-p group is a p-group. (due to J.P. Serre)
See also
