Index of a closed subgroup in a profinite group

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Definition

Suppose G is a profinite group and H is a closed subgroup of G. The index of H in G, in the profinite group sense, is defined as the following supernatural number:

  1. It is the lcm (in the supernatural number sense) of the values of the index [G/U:HU/U] where U varies over the open normal subgroups of G. Note that if U is an open normal subgroup of the profinite group G, then G/U is finite, so the index is actually a finite number.
  2. It is the lcm (in the supernatural number sense) of the values of the index [G:V] where V ranges over the open normal subgroups of G containing H.

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