# Compact and totally disconnected implies every open neighborhood of identity contains an open normal subgroup

From Groupprops

## Statement

Suppose is a profinite group (i.e., it is a compact Hausdorff totally disconnected group) and is an open subset of containing the identity element of . Then, contains an open normal subgroup of .

## Facts used

- Compact neighborhood of identity in totally disconnected group contains compact open subgroup
- Intersection of conjugates by compact subset of open neighborhood of identity contains open neighborhood of identity

## Proof

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