Discussion
Consider the following statements.
I. There exists a proper subgroup \(G\) of \((\mathbb{Q}, +)\) such that \(\mathbb{Q}/G\) is a finite group.
II. There exists a subgroup \(G\) of \((\mathbb{Q}, +)\) such that \(\mathbb{Q}/G\) is isomorphic to \((\mathbb{Z}, +)\).
Which one of the following is correct?
Answer: D
Both I and II are FALSE.
\((\mathbb{Q}, +)\) is divisible: every element can be divided by every positive integer. Any quotient of a divisible group is divisible.
I: a finite divisible group is trivial, since if \(|Q/G| = n\) then every element equals \(n\) times some element, which is 0. So \(\mathbb{Q}/G\) finite forces \(G = \mathbb{Q}\), not a proper subgroup.
II: \((\mathbb{Z}, +)\) is not divisible (1 is not twice an integer), so it cannot be a quotient of \(\mathbb{Q}\). Both statements fail.
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