TITLE: Show that the integers have infinite index in the additive group of rational numbers.
QUESTION [0 upvotes]: I know someone has answered this before but I didn't understand at all what the answer meant so I was hoping someone can answer it again please ?

REPLY [0 votes]: The quotient group $\mathbb{Q}/\mathbb{Z}$ is infinite since for example the cosets $\frac{1}{n} + \mathbb{Z}$ ($n>0$) are pairwise distinct.

REPLY [0 votes]: The set of cosets is precisely the set of all sets of the form $\{k+x|k\in \mathbb Z\}$ where $x$ is of the form $\frac{m}{n}$ where $m$ and $n$ are coprime integers with $m<n$.
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