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Suppose $A = \langle x\mid x^m = e\rangle$, $B = \langle y\mid y^n = e\rangle$, and $C = \langle z\mid z^d = e\rangle$ (these are cyclic groups of order $m,n,d$, respectively). Also suppose $d$ divides $m$ and $n$.

(a) Show that $\phi : C \rightarrow A \times B $, $ z \mapsto (x^{m/d}, y^{n/d})$ is an injective group homomorphism.

(b) Find the elementary divisors of $D = (A \times B)/\phi(C)$ and confirm that $D$ is cyclic iff $d=\mathrm{gcd}(m,n)$.

I think that this problem is related to the Chinese Remainder Theorem and its proof, but I am not sure how to proceed with this. I would appreciate any help with this. Thank you.

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    Thanks for that clarification, @HagenvonEitzen.2012-11-21

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(a) First you have to show that this defines a homomorphism, i.e that $Phi$ respects the relation $z^d=e$ in the presentation of $C$. Indeed, we have $(x^{m/d},y^{n/d})^d = (x^{(m/d)d},y^{(n/d)d})=(x^m,y^n)=(e,e),$ which is th eneutral element of $A\times B$. To show injectivity, assume $z^k\mapsto (e,e)$. That implies $x^{(m/d)k}=e$ and $y^{(n/d)}k=e$ and thus that $\frac md k$ is a multiple of $m$ and $\frac nd k$ a multiple of $n$. Both simply mean that $\frac kd$ is an integer, i.e., $k$ is a multiple of $d$.