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Given the function $f: \frac{\mathbb Z}{mn\mathbb Z} \rightarrow \frac{\mathbb Z}{m\mathbb Z} \times \frac{\mathbb Z}{n\mathbb Z}$ defined by $f([a]_{mn}) = ([a]_m, [a]_n)$.

I must:

Show that $f$ is well-defined.

Show that if $m=6$ and $n=10$ that $f$ is neither injective nor surjective.

Show in general that if $m,n$ have a common divisor $d>1$, then $f$ is neither injective nor surjective.

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    this is famous exercise in abstract algebra2012-04-13

1 Answers 1

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If $[a]_{mn}=[b]_{mn}$ then $mn\ |\ b-a$ so in particular $m | b-a$ and $n|b-a$. Thus the map $f$ is well-defined. Now, if $m=6$ and $n=10$ then $f(30)=0$ but $30\not\equiv 0(mod 60)$ hence $f$ is not injective. On the other hand, since both groups have the same size a function is injective if and only if is surjective. Therefore, $f$ is not surjective too. Using the same idea we can handle the general case.

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    This is what is known as giving a man a fish.2012-04-13