Order of a profinite group

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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

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: