Quotient of subgroups

Question

Is it true that $$\mathbb{Z/4Z\subseteq Z/2Z}$$ Why precisely? Or the reverse $$\mathbb{Z/2Z \subseteq Z/4Z}$$ holds? I'm a beginner. How do I justify the true inclusion? How do I visualize $$\mathbb{Z/2Z \subseteq Z/4Z}$$ Thank you very much.

Answer 1

$\mathbb{Z}/4\mathbb{Z}\subseteq \mathbb{Z}/2\mathbb{Z}$ cannot be true because $\mathbb{Z}/4\mathbb{Z}$ has $4$ elements but $\mathbb{Z}/2\mathbb{Z}$ has only $2$ elements.

$\mathbb{Z}/2\mathbb{Z}\subseteq \mathbb{Z}/4\mathbb{Z}$ is not true because a class mod $2$ is not a class mod $4$.

Nevertheless, the classes $0 + 4\mathbb{Z}$ and $2 + 4\mathbb{Z}$ behave additively like the classes $0 + 2\mathbb{Z}$ and $1 + 2\mathbb{Z}$. In that sense, we may say that $\mathbb{Z}/2\mathbb{Z}\subseteq \mathbb{Z}/4\mathbb{Z}$, but it is a bit of a stretch.

Answer 2

The objects $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ are sets but also they're groups.

Thinking about them as sets we have $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ and $\mathbb{Z}/4\mathbb{Z} = \{0,1,2,3\}$ and as sets $\{0,1\}\subseteq \{0,1,2,3\}$.

This inclusion is not true as groups! It is because $1+1=0$ in $\mathbb{Z}/2\mathbb{Z}$ but $1+1=2$ is $\mathbb{Z}/4\mathbb{Z}$.

It is still possible to think about $\mathbb{Z}/2\mathbb{Z}$ as a subgroup of $\mathbb{Z}/4\mathbb{Z}$ by identifying $1$ with $2$. In other words the subset $\{0,2\}$ under $\mathbb{Z}/4\mathbb{Z}$ addition is "isomorphic" to $\mathbb{Z}/2\mathbb{Z}$ (basically because $2+2=0$ in $\mathbb{Z}/4\mathbb{Z}$).