give the lists of elementary divisors for an abelian group of the specified order and match each list with corresponding list of invariant factors Give the lists of elementary divisors for an abelian group of the order $270$ and match each list with corresponding list of invariant factors.
Why the elementary divisors corresponding to invariant factor $270$ is $2, 27,5$, not $2,9,3,5$ or $2,3,3,3,5$? And why the elementary divisors corresponding to invariant factor $90,3$ is $2, 9,3,5$ and the elementary divisors corresponding to invariant factor $30,3,3$ is $2, 3,3,3,5$?

From the fundamental theorem of abelian groups, there can be exactly $P(n)$ number of non-isomorphic group of order $p^n$, where $P(n)$ is the number of integer partitions of $n$.  
For $n=\prod_{i=0}^n p_i^{n_i}$, where each $p_i$'s are distinct primes, we have $\mathbb{Z}_n=\oplus_{i=0}^n \mathbb{Z}_{p_i^{n_i}}$. Say $A_i$ is the set of all non-isomorphic groups of order $p_i^{n_i}$. Then all the non-isomorphic groups of order $n$ is obtained by taking one group from each $A_i$ and taking their direct sums (note that the order of the direct sums doesn't matter). Thus the total number of non-isomorphic groups of order $n$ is $\prod_{i=0}^nP(n_i)$, $P(n_i)$ being the number of integer partitions of $n_i$
Now, $270 = 2.3^3.5$
The integer partitions of 3 are  


*

*3

*2 + 1

*1 + 1 + 1


So number of non-isomorphic groups of order $270$ is $P(1).P(3).P(1)=1.3.1=3$.
The elementary factors : $(2, 3^3, 5), (2, 3^2, 3, 5), (2, 3, 3, 3, 5)$
i.e any abelian group of order $270$ is isomorphic to one of the following groups :  


*

*$\mathbb{Z}_2\oplus\mathbb{Z}_{3^3}\oplus\mathbb{Z}_5$,  

*$\mathbb{Z}_2\oplus\mathbb{Z}_{3^2}\oplus\mathbb{Z}_3\oplus\mathbb{Z}_5$,  

*$\mathbb{Z}_2\oplus\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_5$  


Now for the invariant factors, using the technique described here, we have the invariant factors : $(270), (3, 90), (3, 3, 30)$
Thus the list of non-isomorphic groups :  


*

*$\mathbb{Z}_{270}$

*$\mathbb{Z}_3\oplus\mathbb{Z}_{90}$

*$\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_{30}$

