# Group of Rationals Modulo One is Group

Jump to navigation
Jump to search

## 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

- 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 $8 \text{(b)}$