The integers as a subgroup of the rationals

#group_theory

The additive group $Z = (Z, +)$ of rational integers is a subgroup of the additive group $Q = (Q, +)$ . Show that $Q$ has infinite index in $Q$ .

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The additive group ${\bf Z}=({\bf Z},+)$ of rational integers is a subgroup of the additive group ${\bf Q}=({\bf Q},+)$. Show that ${\bf Z}$ has infinite index in ${\bf Q}$.