Pro-p group

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In mathematics a pro-p group is a pro-finite group $G$ such that for any open (hence of finite index) normal subgroup $N\triangleleft G$ the quotient group $G / N$ is a finite p-group.

Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of 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 $\mathbb{Q}_p$ 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.