Index of a closed subgroup in a profinite group

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 $H$.

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.