Group of Rationals Modulo One is Group

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Theorem

The set of equivalence classes $\Q/\Z$ with respect to the relation

$a \sim b :\Longleftrightarrow a-b \mathop\in\Z$

with the binary operation

$\Q/\Z \times \Q/\Z \to \Q/\Z, \quad \struct{[a],[b]} \mapsto [a+b]$

is an infinite abelian group.


Proof


Sources