# group,subgroup and isomorphism

I study group theory now but I could not understand isomorphisms very well. In the book that I study I have seen that;

$\mathbb{Z}_6=\{0,1,2,3,4,5\}$ is given and $H=\{0,2\}$ is a subgroup of the group $\mathbb{Z}_6$.

$H=\{0,3\}$ is a isomorphic to the group $\mathbb{Z}_2=\{0,1\}$.

I could not understand this isomorphism part. Why is the set $H$ isomorphic to the set $\mathbb{Z}_2$?

How can we prove that?

• I don't know much about group theory, but I think $H$ is not closed under $+$ or $*$. Is $H = \{0, 3\}$? – JiminP Nov 7 '13 at 13:10
I think you mean, as JiminP pointed out, $H=\{0,3\}$. In order to prove isomorphism, there must be a bijective function from $H$ to $\mathbb{Z}_2$ such that, for $h_1,h_2\in H$ $f(h_1+h_2)=f(h_1)+f(h_2)$. Note that the map $f:\begin{matrix}0\\3\end{matrix}\to\begin{matrix}0\\1\end{matrix}$ satisfies these conditions.
consider $H$ as a group under addition modulo 6, and consider $\mathbb{Z}_2$ which is a group under addition modulo 2. Define a isomorphism by $0 \rightarrow 0$, $3 \rightarrow 1$ . prove that this is an isomorphism of groups. feel the intuition behind the proof by the way map is defined