Pro-p group

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 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).

References

See also

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