If $G= \mathbb{Z}/10\mathbb{Z}$ and $H= 2\mathbb{Z}/10\mathbb{Z}$, then the quotient group $G/H$ is isomorphic to ...

Question

Let $G= \mathbb{Z}/10\mathbb{Z}$. Let $H= 2\mathbb{Z}/10\mathbb{Z}$. Then, the quotient group $G/H$ is isomorphic to ...

1. $\mathbb{Z}/10\mathbb{Z}$
2. $\mathbb{Z}/5\mathbb{Z}$
3. $\mathbb{Z}/2\mathbb{Z}$
4. None of the above

I solved out 3rd option to be the correct answer. Do correct me if I am wrong.

Answer 1

Some points:

• groups $2\mathbb{Z}$ and $10\mathbb{Z}$ are both abelian and of course $10\mathbb{Z}\leq2\mathbb{Z}$. So the group $H$ is defined. We see that $$H=\{\bar0+10\mathbb{Z}, \bar2+10\mathbb{Z},\bar4+10\mathbb{Z},\bar6+10\mathbb{Z},\bar8+10\mathbb{Z}\}\cong\mathbb{Z}_5$$
• With a similar way, you can find $G/H=\dfrac{\mathbb{Z_{10}}}{\mathbb{Z_5}}$.