Non-isomorphic abelian groups of order $3^2\times 5^2\times 7^2$

Question

I just want to make sure that my understanding of the following problem is correct:

Suppose we want to find how many non-isomorphic abelian groups there are of order $3^2\times 5^2\times 7^2$.

Then, since 3,5, and 7 are all prime, we know that the number of non-isomorphic abelian groups of this order will be $2^3=8$, since the number of partitions of 2 is just 2.

Further, if we consider another example, say $3^2\times 5^2\times 7^5$, then we know that there will be (2)(2)(7)=28 non-isomorphic abelian groups of this order because there are 7 partitions of 5. This is done via the Fundamental Theorem for Fin. Gen. Abelian groups.

Is this all correct?

Answer 1

This is correct. Partitions are the key to counting finite abelian groups.