1. ## An isomorphism

Let $(x_1,\ldots,x_r)$ be a non-zero element of $\mathbb{Z}^r$, and let h be the highest common factor of $x_1,\ldots,x_r$. Show that there is an isomorphism $\mathbb{Z}^r \to \mathbb{Z}^r$ taking $(x_1,\ldots,x_r)$ to (1, 0, 0,..., 0) if and only if h = 1.

Deduce that $\mathbb{Z}^r / \langle (x_1,\ldots,x_r)\rangle \cong \mathbb{Z}^{r-1} \oplus (\mathbb{Z}/h\mathbb{Z})$.
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The first part isn't a problem. But the deduction is something I'm having trouble with... help?

2. Originally Posted by Capillarian
Let $(x_1,\ldots,x_r)$ be a non-zero element of $\mathbb{Z}^r$, and let h be the highest common factor of $x_1,\ldots,x_r$. Show that there is an isomorphism $\mathbb{Z}^r \to \mathbb{Z}^r$ taking $(x_1,\ldots,x_r)$ to (1, 0, 0,..., 0) if and only if h = 1.

Deduce that $\mathbb{Z}^r / \langle (x_1,\ldots,x_r)\rangle \cong \mathbb{Z}^{r-1} \oplus (\mathbb{Z}/h\mathbb{Z})$.
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The first part isn't a problem. But the deduction is something I'm having trouble with... help?
I like this question. It is subtle. So, you have proven the first part. Now, can you use this to generalise the result? That is, use it to prove that there is an isomorphism $\phi:\mathbb{Z}^r \rightarrow \mathbb{Z}^r$ with $(x_1, \ldots, x_r)\mapsto (h, 0, 0, \ldots, 0)$ where $h=gcd(x_1, x_2, \ldots, x_r)$.

HINT: This isomorphism is the same as the one you've found already...

Can you see how what you want follows from this result?

3. Originally Posted by Swlabr
I like this question. It is subtle. So, you have proven the first part. Now, can you use this to generalise the result? That is, use it to prove that there is an isomorphism $\phi:\mathbb{Z}^r \rightarrow \mathbb{Z}^r$ with $(x_1, \ldots, x_r)\mapsto (h, 0, 0, \ldots, 0)$ where $h=gcd(x_1, x_2, \ldots, x_r)$.

HINT: This isomorphism is the same as the one you've found already...

Can you see how what you want follows from this result?
Ah, then I think I've made a mistake in the proof of the first part. The ' $\implies$' direction still works, but if h = 1, how do you go about choosing an explicit isomorphism that maps $(x_1, \ldots, x_r)\mapsto (1, 0, 0, \ldots, 0)$?

4. The isomorphism is basically just $(x_1, x_2, \ldots, x_n) \mapsto (x_1i_1+x_2i_r \dlots x_ni_n, 0, \ldots, 0)$ where $x_1i_1+x_2i_2+\ldots+x_ni_n=gcd(x_j)=1$. This is an isomorphism as one can easily prove that the $i_j$ are coprime (I'll leave you to do this) and so they `act like' primes. Again, I will leave you to prove that this is an isomorphism.