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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.

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2 Answers 2

up vote 8 down vote accepted

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.

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

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I don't quite understand this. Could you please give some more detail or an example? Thank you. –  user49097 Nov 21 '12 at 20:36

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