Show that each subgroup generated by prime integer is maximal in $(\Bbb Z, +)$.

Question

Show that each subgroup generated by prime integer is maximal in $(\Bbb Z, +)$.

Here I know that we can prove maximal by showing its quotient group is simple.

But how can I approach "each subgroup generated by prime integer"?

Answer 1

So if you work out as Shaun mentioned in the comment, you will see that the subgroup generated by the prime number $p $, $$\langle p \rangle=\{\ldots,-2p,-p,0,p,\ldots\}=p\Bbb Z.$$

In fact,since $\Bbb Z$ is cyclic, every subgroup of $\Bbb Z$ is cyclic,i.e.,every subgroup of $\Bbb Z$ is of the form $n\Bbb Z$ for some $n\in \Bbb Z$.

To show $p\Bbb Z$ is maximal in $\Bbb Z$, you can use the fact that$$\frac{\Bbb Z}{p\Bbb Z}\cong \Bbb Z_p. $$