Pro-p group

In mathematics, a pro-p group (for some prime number p) is a profinite group $G$ such that for any open normal subgroup $N\triangleleft G$ 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 $\mathbb{Q}_p$ 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 $r$ such that any closed subgroup has a topological generating set with no more than $r$ elements.

Examples

The canonical example is the p-adic integers

$\mathbb{Z}_{p} = \displaystyle \lim_{\leftarrow} \mathbb{Z}/p^n\mathbb{Z}.$

The group $\ GL_{n}( \mathbb{Z}_{p})$ of invertible n by n matrices over $\ \mathbb{Z}_{p}$ has an open subgroup U consisting of all matrices congruent to the identity matrix modulo $\ p\mathbb{Z}_{p}$ . This U is a pro-p group. In fact the p-adic analytic groups mentioned above can all be found as closed subgroups of $\ GL_{n}( \mathbb{Z}_{p})$ for some integer n,
Any finite p-group is also a pro-p-group (with respect to the constant inverse system).

References

Dixon, J. D.; du Sautoy, M. P. F.; Mann, A.; Segal, D. (1991), Analytic pro-p-groups, Cambridge University Press, ISBN 0-521-39580-1, MR 1152800

Pro-p group

Examples

References

See also