Why is this function not a group isomorphism?

Question

I have defined a homorphism $\phi: Z/100Z * Z/10Z \rightarrow Z/1000Z$ defined as $\phi ((a,b)) \rightarrow a*10 + b$ and from what I can tell this is a valid isomorphism. Can someone explain to me why my function is not an isomorphism?

edit: reworded to show uniqueness of question

Answer 1

\begin{align} \phi\big((0,5)\big) &= 5, \\ (0,5)+(0,5) = (0,5+5)&=(0,0),\qquad\text{the second coordinate computed in }Z/10Z \\ \phi\big((0,5)+(0,5)\big) = \phi\big((0,0)\big) &= 0 \ne 10 = 5+5 \qquad\text{the $5+5$ computed in }Z/1000Z \end{align}

Answer 2

Here is one reason (among many, but you only need one) why your function is not an isomorphism.

In $\mathbb Z / 1000 \mathbb Z$ we have $\phi(100,10) = 100*10 + 10 = 1010 = 10$ modulo $1000$.

But $100 = 0$ mod $100$, and $10 = 0$ mod $10$, so $\phi(100) * \phi(10) = \phi(0) * \phi(0) = 0$ mod $1000$.

And $10 \ne 0$ modulo $1000$.