Here is a question which I came up with.
Let be a finite non-abelian group. What are all the possible cardinalities of this group?
For example, it cannot be 1 because that is the trivial group which is abelian. It cannot be 2 or 3, because up to isomorphism there is only 1 namely which is abelian. With 4 you have two possibilities the usual and the Klein 4 group which is also abelian. With 5 (prime) there can only be . Now with 6 we finally have our first non-abelian group, the dihedral group. With 7 since it is prime we have the same thing ...
Which orders are possible?