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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"?

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No, they are not the same. You can tell this from the theorem that you stated: it clearly indicates that a finitely generated Abelian group can be infinite. In particular, $\Bbb Z$ under addition is an infinite, finitely generated Abelian group. However, every finite Abelian group is finitely generated, so the theorem applies to finite Abelian groups. –  Brian M. Scott Mar 10 '12 at 12:45
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No, they are not. A finite abelian group is also finitely generated. However a finitely generated abelian group need not be finite. Look at $\mathbb Z$ as a group under addition.

This group is generated by a single element, say $1$ or $-1$. However, this is not finite.

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To expand on my comment, all finite Abelian groups are finitely generated, but not all finitely generated Abelian groups are finite. An Abelian group $\langle G,+\rangle$ is finitely generated if there is a finite $F\subseteq G$ such that every element of $G$ can be written as the sum of elements of $F$ or their inverses, possibly with repetition. Thus, $\langle\Bbb Z,+\rangle$ is finitely generated: we may take $F=\{1\}$, since every integer is the sum of finitely many copies of $1$ or its inverse $-1$.

Every finite Abelian group is clearly finitely generated, so the structure theorem that you quote applies to them. In particular, it says that if $G$ is a finite Abelian group, then it must be the direct sum of cyclic groups whose orders are powers of primes. That is the conclusion that is being used in the particular example that you’re looking at.

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