I'm a little confused. In my textbook, they try to determine the abelian groups of order $1500 = 2^{2}\cdot 3\cdot 5^{3}$. They find the following families of elementary divisors: $\{2,2,3,5^{3}\}$ , $\{2,2,3,5,5^{2}\}$, $\{2,2,3,5,5,5\}$, $\{2^{2},3,5^{3}\}$, $\{2^{2},3,5,5^{2}\}$ and $\{2^{2},3,5,5,5\},$ and then argue that each of these determines an abelian group of order 1500 and that every abelian group of order 1500 is isomorphic to one of these. In order to deduce that every abelian group of order 1500 is isomorphic to one of these, they use following theorem:
Every finitely generated abelian group $G$ is (isomorphic to) a finite direct sum of cyclic groups, each of which is either infinite or of order a power of a prime.
My problem is that the theorem is about finitely generated abelian groups and not "just" finite abelian groups. Are a finitely generated abelian group and a finite abelian group the same "thing"?