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 N\triangleleft G the quotient group G/N 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 {\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

{\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).

See also

References


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