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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.
== Examples ==
Important examples of locally profinite groups come from algebraic number theory. Let ''F'' be a non-archimedean local field. Then both ''F'' and are locally profinite. More generally, the matrix ring and the general linear group 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).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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