Show that $4ℤ/12ℤ ≃ ℤ_3$

Question

I am trying to understand how $4ℤ/12ℤ$ is isomorphic to $ℤ_3$. So far I understand that: $$4ℤ/12ℤ = \{ 0+12ℤ,4+12ℤ,8+12ℤ \}$$ and that elements of integer $\bmod 3$ are: $$ℤ_3 =\{ 0,1,2\}$$ However, I cannot figure out how $4ℤ/12ℤ$ is isomorphic to $ℤ_3$. Any help would be very much appreciated.

Answer 1

One possibility to solving your problem would be by constructing an isomorphism between the two groups. For example consider: $$ \varphi: 4\mathbb{Z} /12\mathbb{Z} \to \mathbb{Z} / 3 \mathbb{Z} $$ $$ 0+12z \mapsto 0+12z \mod 3, \qquad 4+12z \mapsto 4+12z \mod 3, \qquad 8+12z \mapsto 8+12z \mod 3 $$ (So really $[0] \mapsto [0]$, $[4] \mapsto [1]$ and $[8] \mapsto [2]$)

One would now only have to prove that this is indeed an group homomorphism ($\varphi(a+b) = \varphi(a) + \varphi(b)$) and especially isomorphic.

Also one would have to prove the independence of the homomorphism of the variable $z$, which is somewhat obvious since $12z \equiv 0 \mod 3$ for all $z$.