# Some confusion on an example of Hungerford's book

Here is an example on Hungerford's Algebra, page 31.

But I don't think this argument is correct, according to the definition of homomorphism, $g(xy)=g(x)g(y)~x,y\in \mathbf{Z}_m$ that means the congruence equation $$k(xy\bmod{m}) \equiv (kx)(ky)\pmod{km}$$but the preceding equation may not be true for an arbitrary positive integer $k$.

Did I have any misunderstanding on this example?

• The operation is addition, not multiplication. Sep 19, 2016 at 10:34
• Just to emphasize: $\mathbb Z_m$ is not even a group under multiplication; $0$ has no inverse.
– lulu
Sep 19, 2016 at 10:36

$Z_m$ is not always a group under multiplication. So when he is referring to the map
$g: Z_m \to Z_{mk}$ $x \mapsto kx$
then $g(x+y) = k(x+y)=kx+ky=g(x)+g(y)$
thus it's a homomorphism. As to why is injectiv just note, that the image has the same number of elements as $Z_m$ has.