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
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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 $8 \text{(b)}$