# Lagrange's theorem for profinite groups

From Groupprops

## Statement

Suppose is a profinite group and is a closed subgroup of . Then, we have:

where:

- denotes the order of in the sense of order of a profinite group, and is a supernatural number.
- denotes the order of in the sense of order of a profinite group, and is a supernatural number.
- denotes the index of in in the sense of index of a closed subgroup in a profinite group, and is a supernatural number
- The multiplication on the right is in the sense of multiplication of supernatural numbers.

## Related facts

- Index is multiplicative for profinite groups
- Lagrange's theorem is the ordinary version.