I'm trying to understand why, in Example 6.7 (pg 239) of Cracking the GRE Math, we know that each unique list of elementary divisors gives rise to an abelian group that is non-isomorphic to the other groups defined by the list.

The groups are certainly the same order...

I understand that 6) is a black sheep because it is the only cyclic group (gcd of the m's is 1).

But what about the others? How do we know that they are mutually non-isomorphic?

btw, to write the above in tex, alls I typed was Z_8 \oplus Z_3 \oplus Z_{25}