Let $k$ be the homomorphism from $\mathbb Z$ to $\mathbb Z/2\mathbb Z\times\mathbb Z$ defined by $1 \mapsto (1 + 2\mathbb Z, -2)$. What is the quotient group of $\mathbb Z/2\mathbb Z\times\mathbb Z$ modulo the image of $k$?
I first thought it would be easy... But it turns out to be quite nontrivial for me. Could anyone help me?
(Answering for my own practice. If someone sees this and lets me know if I am correct, much appreciated.)
The image of $k$ is isomorphic to $\mathbb{Z}_2 \times 2\mathbb{Z}$. Since $(A \times B) / (H\times K) \cong A/H \times B/K$ if $H \unlhd A$ and $K \unlhd B$, we get, $$ \mathbb{Z}_2 \times 2\mathbb{Z} / \text{im}(k) \cong \{0\} \times \mathbb{Z}_2 \cong \mathbb{Z}_2 $$