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If there

There are exactly two at most 2 groups (up to isomorphism) of order $n$, then and there is not a field of order $n$, if and only if $n$ is the product of two distinct primes.

This isn't if and only if, it doesn't capture "distinct", and it doesn't cover "prime powers", but at least it's nontrivial.

1

If there are exactly two groups (up to isomorphism) of order $n$, then $n$ is the product of two primes.

This isn't if and only if, it doesn't capture "distinct", and it doesn't cover "prime powers", but at least it's nontrivial.