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.


2 Answers 2



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.


Express each factor as the product of maximal prime power subgroups, but remember not to combine these between factors.

  • $\begingroup$ I don't quite understand this. Could you please give some more detail or an example? Thank you. $\endgroup$
    – user49097
    Commented Nov 21, 2012 at 20:36

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .