Locally profinite group

In mathematics, a locally profinite group is a hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is hausdorff locally compact and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and p-adic Lie group. Non-examples are real Lie groups which have no small subgroup property.

In a locally profinite group, a closed subgroup is locally profinite, and every compact subgroup is contained in an open compact subgroup.

Important examples of locally profinite groups come from algebraic number theory. Let F be a non-archimedean local field. Then both F and ${\displaystyle F^{\times }}$ are locally profinite. More generally, the matrix ring ${\displaystyle \operatorname {M} _{n}(F)}$ and the general linear group ${\displaystyle \operatorname {GL} _{n}(F)}$ are locally profinite. Another example of a locally profinite group is the absolute Weil group of a non-archimedean local field: this is in contrast to the fact that the absolute Galois group of such is profinite (in particular compact).

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