Use the first isomorphism theorem to show that $(\mathbb Z \times \mathbb Z \times \mathbb Z)/\langle(4, 4, 4)\rangle$ is isomorphic to $\mathbb Z \times \mathbb Z \times \mathbb Z_4$.
I´m a newbie in abstract algebra, and I find the course very hard. Would be nice if someone could explain this in an easy way (if possible) :)
"Use the first isomorphism theorem ..." means we're supposed to find a surjective group homomorphism $\varphi\colon \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ whose kernel is the cycling subgroup generated by $(4,4,4)$. In other words, we have to have $\varphi(a,b,c) = 0$ if and only if $a=b=c\equiv 0 \text{ (mod }4\text{)}$. Obviously, $\varphi(a,b,c) := (a-b,b-c,c+4\mathbb{Z})$ does have this property; I'll leave it up to you to check that this really defines a surjective group homomorphism.