TITLE: Show that 2Z and 3Z are not isomorphic - question on proof
QUESTION [6 upvotes]: I need to show that $2\Bbb Z$ and $3\Bbb Z$ are not isomorphic.
I found a contradiction as follows: let $p$ be this isomorphism from $2\Bbb Z$ to $3\Bbb Z$.
Then $p(4) = p(2*2) = p(2+2)$, so
$p(2)p(2) = p(2)+p(2)$.
Since $p(2)$ is in $3\Bbb Z$, let us call it $3k$ where $k$ is an integer.
So
$9k² = 6k$
$k = 2/3$
This is a contradiction because $k$ is an integer.
My confusion here is that this does not rely on the bijectivity of p! Where does this fail for a homomorphism?

REPLY [15 votes]: $9 k^2 = 6 k$ (for integer $k$) if and only if $k = 0$.  Now here's the point where you use bijectivity: if $k = 0$, $p(2) = 0$ so $p$ would not be one-to-one.