# Index of a closed subgroup in a profinite group

From Groupprops

## Definition

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

- It is the lcm (in the supernatural number sense) of the values of the index where varies over the open normal subgroups of . Note that if is an open normal subgroup of the profinite group , then is finite, so the index is actually a finite number.
- It is the lcm (in the supernatural number sense) of the values of the index where ranges over the open normal subgroups of containing .

## 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.