Question

Which one of the following statements is true?

• A. \((\mathbb{Z}, +)\) is isomorphic to \((\mathbb{R}, +).\)
• B. \((\mathbb{Z}, +)\) is isomorphic to \((\mathbb{Q}, +).\)
• C. \((\mathbb{Q}/\mathbb{Z}, +)\) is isomorphic to \((\mathbb{Q}/2\mathbb{Z}, +).\)
• D. \((\mathbb{Q}/\mathbb{Z}, +)\) is isomorphic to \((\mathbb{Q}, +).\)

Hint:

Quotient groups like \(\mathbb{Q}/\mathbb{Z}\) represent rational numbers modulo integers, forming torsion groups — useful in group theory and number theory.

Answer 1

The Correct Option is C

Step 1: Analyze each group.
\((\mathbb{Z},+)\) is a discrete infinite cyclic group. \((\mathbb{R},+)\) and \((\mathbb{Q},+)\) are uncountable and countable divisible groups respectively — hence not isomorphic to \(\mathbb{Z}\).
Step 2: Compare quotient groups.
\(\mathbb{Q}/\mathbb{Z}\) consists of all fractional parts of rationals — it is a torsion group (every element has finite order). Similarly, \(\mathbb{Q}/2\mathbb{Z}\) is obtained by modding out by \(2\mathbb{Z}\), which preserves the torsion property and structure. Hence, they are isomorphic.
Step 3: Conclusion.
Therefore, (C) is true.