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Show that Z₁ Z18. From an abstract algebra, senior level university math course

Show that Z₁ Z18. From an abstract algebra, senior level university math course

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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**Title: Isomorphism Between Units of Modular Integers** **Problem Statement**: Show that \( \mathbb{Z}_{14}^* \cong \mathbb{Z}_{18}^* \). **Course Context**: From an abstract algebra, senior level university math course. --- To solve this problem, one must establish a group isomorphism between the units of the integers modulo 14 and modulo 18. The units, \( \mathbb{Z}_n^* \), are the elements that have a multiplicative inverse under modulo \( n \). **Detailed Steps**: 1. **Identify Units**: - Calculate the units of \( \mathbb{Z}_{14} \). - Calculate the units of \( \mathbb{Z}_{18} \). 2. **Verify Group Properties**: - Ensure both sets of units form groups under multiplication. 3. **Establish an Isomorphic Mapping**: - Define a bijective homomorphism between \( \mathbb{Z}_{14}^* \) and \( \mathbb{Z}_{18}^* \). 4. **Show Bijectiveness and Homomorphism**: - Prove that the map is both injective and surjective. - Verify that the homomorphism respects the group operation. By confirming each of these elements, one can demonstrate that \( \mathbb{Z}_{14}^* \) and \( \mathbb{Z}_{18}^* \) are isomorphic groups. This result underscores the intrinsic symmetry between these modular integer structures in the context of group theory. This exercise illustrates fundamental concepts in abstract algebra, particularly the properties and applications of group theory in understanding isomorphic structures.
Transcribed Image Text:**Title: Isomorphism Between Units of Modular Integers** **Problem Statement**: Show that \( \mathbb{Z}_{14}^* \cong \mathbb{Z}_{18}^* \). **Course Context**: From an abstract algebra, senior level university math course. --- To solve this problem, one must establish a group isomorphism between the units of the integers modulo 14 and modulo 18. The units, \( \mathbb{Z}_n^* \), are the elements that have a multiplicative inverse under modulo \( n \). **Detailed Steps**: 1. **Identify Units**: - Calculate the units of \( \mathbb{Z}_{14} \). - Calculate the units of \( \mathbb{Z}_{18} \). 2. **Verify Group Properties**: - Ensure both sets of units form groups under multiplication. 3. **Establish an Isomorphic Mapping**: - Define a bijective homomorphism between \( \mathbb{Z}_{14}^* \) and \( \mathbb{Z}_{18}^* \). 4. **Show Bijectiveness and Homomorphism**: - Prove that the map is both injective and surjective. - Verify that the homomorphism respects the group operation. By confirming each of these elements, one can demonstrate that \( \mathbb{Z}_{14}^* \) and \( \mathbb{Z}_{18}^* \) are isomorphic groups. This result underscores the intrinsic symmetry between these modular integer structures in the context of group theory. This exercise illustrates fundamental concepts in abstract algebra, particularly the properties and applications of group theory in understanding isomorphic structures.
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