The order of elements in infinite quotient groups There are two statements that my professor made today that I'm hoping I can get some more clarification on.
The first is that $\mathbb{Q}/\mathbb{Z}$ is an infinite quotient group where every element has finite order.
The second is that $\mathbb{R}/ \mathbb{Q}$ is also an infinite quotient group but every element except the identity has infinite order.
I'm having trouble even imagining an infinite quotient group...I'm familiar with groups like $\mathbb{Z} / n\mathbb{Z}$, but how would you even notate these other groups? I understand that in order for an element $xH$, where $H$ is the subgroup, to have finite order, $x^n$ must be in $H$ for some $n$. So if every element of $\mathbb{Q}/\mathbb{Z}$ is finite, does that imply that every rational number is in $\mathbb{Z}$? Obviously that is not true, but I'm having trouble figuring out where I'm going wrong.

You have to remember what the operation is. The notation $x^n$ is used in general groups. However, if $\mathbb{Q}$ and $\mathbb{Z}$ then the operation is addition, not multiplication. So every element in $\mathbb{Q}/\mathbb{Z}$ having a finite order means that for every rational number $a\in\mathbb{Q}$ there is some $n\in\mathbb{N}$ such that $na=a+...+a\in\mathbb{Z}$. And this is indeed true. If $a=\frac{p}{q}$ for $p\in\mathbb{Z}$ and $q\in\mathbb{N}$ then $qa\in\mathbb{Z}$.
