Explain why the correspondence $x \mapsto 3x$ from $\Bbb Z_{12}$ to $\Bbb Z_{10}$ is not a homomorphism.

Question

Explain why the correspondence $x \mapsto 3x$ from $\Bbb Z_{12}$ to $\Bbb Z_{10}$ is not a homomorphism.

Here image of $1$ is $3$ and $|3| = 10 $ which does not divide $12$, the order of $1$. So this can't be homomorphism.

Is this correct?

Answer 1

This "map" is doomed from the outset, as it is not well defined.

As it fails to be a bonified function, it certainly can't be a homomorphism.

Besides, it is easy to see the only homomorphism, other than the trivial one, from $\Bbb Z_{12}$ to $\Bbb Z_{10}$ is $h$, given by $h(1)=5$.

Answer 2

Yes, your argument looks good.

Call your map $f$. On the one hand $f(0)= f(12\cdot 1)$. Therefore, for $f$ to be a homomorphism, one must also have $$ 0 = f( 0 ) = f(12\cdot 1) = 12 \cdot f(1) .$$ So (a factor of) $12$ would have to kill $f(1)$ in ${\mathbb Z}_{10}$ - but $f(1)=3$, and $$ 12 \cdot 3 = 6 \not= 0\in {\mathbb Z}_{10}.$$ So no factor of $12$ kills $f(1)$.