Order of a profinite group

Definition
Suppose $$G$$ is a defining ingredient::profinite group. The order of $$G$$ is a defining ingredient::supernatural number defined in the following equivalent ways:


 * 1) 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 $$G$$.
 * 2) It is the lcm (in the supernatural number sense) of the orders of all the finite groups arising as quotients of $$G$$ by some open normal subgroup of it.

Related notions

 * Exponent of a profinite group
 * Index of a closed subgroup in a profinite group

Relation with usual notion of order
If $$G$$ happens to be a finite group, this coincides with the usual notion of order of a group. If $$G$$ 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:


 * Order of a profinite group need not determine order as a group in the sense of cardinality of underlying set
 * Cardinality of underlying set of a profinite group need not determine order as a profinite group