Is it true that $(\mathbb{Z}\bigoplus\mathbb{Z}\bigoplus\mathbb{Z})/(\mathbb{Z}\bigoplus\mathbb{Z})\cong\mathbb{Z}$

Question

Can anyone help? I am interested and very new in studying homology groups, that are correlated to the study of free abelian groups, i.e. groups of form $\mathbb{Z}\bigoplus\mathbb{Z}\bigoplus...\bigoplus\mathbb{Z}$, so I would like to know the behaviour of those groups when it comes to quotient groups.

Answer 1

What you've written isn't true because $\mathbb{Z} \oplus \mathbb{Z}$ isn't a subgroup of $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$. However, assuming you meant to ask whether (for example) $(\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}) / (0 \oplus \mathbb{Z} \oplus \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, then this is true. To see this, define a map $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$ by $(x,y,z) \mapsto x$. This has kernel $(0 \oplus \mathbb{Z} \oplus \mathbb{Z})$, and the first isomorphism theorem does the rest.

Answer 2

The group $\Bbb Z^3$ has many subgroups $A$ isomorphic to $\Bbb Z^2$. The quotient $\Bbb Z^3/A$ can have many structures: it can be isomorphic to $\Bbb Z\oplus \Bbb Z/a\Bbb Z\oplus\Bbb Z/b\Bbb Z$ for any positive integers $a$ and $b$.