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.

Mark's comment explains the case of $\mathbb{Q}/\mathbb{Z}$. For the case $\mathbb{R}/\mathbb{Q}$, you just have to note that every nontrivial element of $\mathbb{R}/\mathbb{Q}$ is of the form $x+\mathbb{Q}$ where $x$ is an irrational number. Thus, if there exists a positive integer $n$ such that $(x+\mathbb{Q})^{n}=nx+\mathbb{Q}=0$, then there exist $a,b\in\mathbb{Z}$, $b\neq0$ such that $nx=\frac{a}{b}$. But then we get $x=\frac{a}{nb}\in\mathbb{Q}$, a contradiction. Therefore, every nontrivial element of $\mathbb{R}/\mathbb{Q}$ has infinite order.
