Hw11 Problem 2

### Problem 2

Classify the group $(\mathbb{Z} × \mathbb{Z} × \mathbb{Z}_8)/\langle(0, 4, 0)\rangle$ according to the fundamental theorem of finitely generated abelian groups.

Solution

We conjecture that $(\mathbb{Z} × \mathbb{Z} × \mathbb{Z}_8)/\langle(0, 4, 0)\rangle$ is isomorphic to $\mathbb{Z} × \mathbb{Z}_4 × \mathbb{Z}_8$, because only the multiples of 4 in the second factor are collapsed to zero. It is easy to check that $\phi: \mathbb{Z} × \mathbb{Z} × \mathbb{Z}_8 \rightarrow \mathbb{Z} × \mathbb{Z}_4 × \mathbb{Z}_8$ defined by $\phi(n, m, s) = (n, r, s)$, where r is the remainder of m when divided by 4 in the division algorithm, is an onto homomorphism with kernel $\langle(0, 4, 0)\rangle$.