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

## Statement

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.

## Related facts

## Proof

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.