Isomorphisms between finite abelian groups I am working on the following problem:
Are there any isomorphisms between the following finite abelian groups?:
$$\mathbb{Z}_{1225}\times\mathbb{Z}_{315},$$  $$\mathbb{Z}_{1575}\times\mathbb{Z}_{245},$$  $$\mathbb{Z}_{75}\times\mathbb{Z}_{49} \times \mathbb{Z}_{105},$$ $$\mathbb{Z}_{175}\times\mathbb{Z}_{45} \times \mathbb{Z}_{49}$$
I know that the solution requires the structure theorem for finite abelian groups and factoring the group orders into elementary divisors, but I am not sure how to proceed using this strategy.  I would greatly appreciate help with this.  Thank you.
 A: HINTS: 
You'll want to decompose the groups, where appropriate. 
Definition: A group $G$ is decomposable if (and only if) it is isomorphic to a direct product of two proper subgroups. Otherwise, $G$ is indecomposable. 


*

*Note, for example, that from this definition it follows that the factor $\mathbb{Z}_{49} = \mathbb{Z}_{7^2}$ in the third and fourth groups is indecomposable. 


In fact, the finite indecomposable groups are exactly the cyclic groups with order a power of a prime.
Recall that $\mathbb{Z}_m \times \mathbb{Z}_n \cong \mathbb{Z}_{mn}$ if and only if $\text{ gcd}(m, n)=1$.


*

*For example, $\mathbb{Z}_5 \times \mathbb{Z}_9 \cong \mathbb{Z}_{45}$. Note that $\text{gcd}(5, 9) = 1.$ 


More generally, the group $$\prod_{i=1}^n \mathbb{Z}_{m_i} \cong \mathbb{Z}_{m_1m_2\cdots m_n}$$ if and only if the numbers $m_i,\;\text{for}\;i = 1, 2, \cdots n$ are pairwise relatively prime (that is, if and only if for each $m_i, m_j,\; 1 \leq i \leq n,\; 1\leq j\leq n,\; i\neq j,\;\text{ gcd}(m_i, m_j) = 1$.) 

Take, as an example, the last group: $$\mathbb{Z}_{175}\times\mathbb{Z}_{45} \times \mathbb{Z}_{49}$$
To decompose this group, note:


*

*$\mathbb{Z}_{175} \cong \mathbb{Z}_{25} \times \mathbb{Z}_7 = \mathbb{Z}_{5^2} \times \mathbb{Z}_7$

*$\mathbb{Z}_{45} \cong \mathbb{Z}_{3^2} \times \mathbb{Z}_5$,

*$\mathbb{Z}_{49} \cong \mathbb{Z}_{7^2}$


Rearranging factors gives us: $$\mathbb{Z}_{175}\times\mathbb{Z}_{45} \times \mathbb{Z}_{49}\cong \mathbb{Z}_{3^2} \times \mathbb{Z}_5 \times \mathbb{Z}_{5^2} \times \mathbb{Z}_7 \times \mathbb{Z}_{7^2}$$
Doing this with each of the groups you list will make it clear which group(s) are isomorphic to which groups.
A: Express each factor as the product of maximal prime power subgroups, but remember not to combine these between factors.
