# Order of a profinite group

From Groupprops

## Definition

Suppose is a profinite group. The **order** of is a supernatural number defined in the following equivalent ways:

- It is the lcm (in the supernatural number sense) of the orders of all the finite groups in some choice of inverse system of finite discrete groups whose inverse limit is .
- It is the lcm (in the supernatural number sense) of the orders of all the finite groups arising as quotients of by some open normal subgroup of it.

## Related notions

## Relation with usual notion of order

If happens to be a finite group, this coincides with the usual notion of order of a group. If is infinite, the order as a profinite group is distinct from the usual notion of order, which is the cardinality of the underlying set. In fact, neither order value can be deduced from the other. See: