Index of a closed subgroup in a profinite group

Definition
Suppose $$G$$ is a defining ingredient::profinite group and $$H$$ is a defining ingredient::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 $$H$$.

Related notions

 * Order of a profinite group is simply the index of the trivial subgroup in it.
 * Index of a subgroup is the usual definition in terms of the cardinality of the left coset space. For a subgroup of finite index, the two definitions coincide.