It doesn't have much to do with how "large" the set is. The complex numbers have the same cardinality as the reals; the rational numbers have the same cardinali

It doesn't have much to do with how "large" the set is. The complex numbers have the same cardinality as the reals; the rational numbers have the same cardinality as the integers and whole numbers. Even talking about finite sets, the group of integers modulo 13 is a lot less "interesting" than the group of integers modulo 12, because how such a group acts depends heavily on the division structure of the modulus: the more divisors there are in the modulus, the more interesting things get. So the sets of integers modulo any prime are all basically the same (and boring), while the integers modulo highly composite numbers like 60 are wild, complex structures.