Pro-p group
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In mathematics, a pro-p group (for some prime number p) is a profinite group G such that for any open normal subgroup 
the quotient group G / N is a p-group. Note that, as profinite groups are compact, the open subgroup must be of finite index, so that the discrete quotient group is 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 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 r such that any closed subgroup has a topological generating set with no more than r elements.
[edit] Examples
- The canonical example is the p-adic integers,
: the inverse limit of the system 
. - The group
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.
[edit] References
- Dixon, J. D.; du Sautoy, M. P. F.; Mann, A. & Segal, D. (1991), Analytic pro-p-groups, Cambridge University Press, MR1152800, ISBN 0-521-39580-1
