Order of a profinite group need not determine order as a group in the sense of cardinality of underlying set
It is possible to have two profinite groups and that have the same order as each other in the sense of order of a profinite group (note that both order numbers are supernatural numbers), but that have different orders from each other in the sense of the cardinality of the underlying set.
Fix a nontrivial finite group . Pick two infinite cardinals such that the power cardinals of are not equal. Now, consider the external direct powers (repeated unrestricted external direct product of with itself): and , both equipped with the product topology from the discrete topology on . We note that:
- For both groups, the order in the sense of a profinite group is as follows: all primes that divide the order of occur with a power of , but no other primes occur. In particular, the order of equals the order of in the sense of order as a profinite group, where the equality is as equality of supernatural numbers.
- The cardinality of the underlying set of is , which is the power cardinal of (assuming the axiom of choice). The cardinality of the underlying set of is , which is the power cardinality of (assuming the axiom of choice). By assumption, these two power cardinals are distinct.