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13. Prove or disprove: Q/ZQ.

13. Prove or disprove: Q/ZQ.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**13. Prove or disprove: \(\mathbb{Q}/\mathbb{Z} \cong \mathbb{Q}\).** The problem asks if the quotient group \(\mathbb{Q}/\mathbb{Z}\) is isomorphic to the group \(\mathbb{Q}\). To explore this, consider the following: - \(\mathbb{Q}\) represents the set of all rational numbers. - \(\mathbb{Z}\) is the set of integers. - The quotient group \(\mathbb{Q}/\mathbb{Z}\) consists of the equivalence classes of rational numbers, where two rationals are considered equivalent if their difference is an integer. The problem involves determining whether there exists a bijective group homomorphism between \(\mathbb{Q}/\mathbb{Z}\) and \(\mathbb{Q}\). This requires proving or disproving the existence of such a mapping that preserves group operations.
Transcribed Image Text:**13. Prove or disprove: \(\mathbb{Q}/\mathbb{Z} \cong \mathbb{Q}\).** The problem asks if the quotient group \(\mathbb{Q}/\mathbb{Z}\) is isomorphic to the group \(\mathbb{Q}\). To explore this, consider the following: - \(\mathbb{Q}\) represents the set of all rational numbers. - \(\mathbb{Z}\) is the set of integers. - The quotient group \(\mathbb{Q}/\mathbb{Z}\) consists of the equivalence classes of rational numbers, where two rationals are considered equivalent if their difference is an integer. The problem involves determining whether there exists a bijective group homomorphism between \(\mathbb{Q}/\mathbb{Z}\) and \(\mathbb{Q}\). This requires proving or disproving the existence of such a mapping that preserves group operations.
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