Group of Rationals Modulo One is Group

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

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Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
• 1974: Thomas W. Hungerford: Algebra ... (previous): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $8 \text{(b)}$