Order of a profinite group

Definition

Suppose $G$ is a profinite group. The order of $G$ is a 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