Answered: Prove that Zo is not isomorphic to Z,…

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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Prove this WITHOUT using the Primitive Root Theorem

$### Title: Proving Non-Isomorphism Between $ \mathbb{Z}_{10}^{\times} $ and $ \mathbb{Z}_{12}^{\times} $ **Problem Statement:** Prove that $ \mathbb{Z}_{10}^{\times} $ is not isomorphic to $ \mathbb{Z}_{12}^{\times} $. **Explanation:** In this problem, we are tasked with proving that the group of units modulo 10, denoted as $ \mathbb{Z}_{10}^{\times} $, is not isomorphic to the group of units modulo 12, denoted as $ \mathbb{Z}_{12}^{\times} $. Here $ \mathbb{Z}_n^{\times} $ stands for the multiplicative group of integers modulo $ n $. An *isomorphism* between two groups is a bijective (one-to-one and onto) group homomorphism. For two groups to be isomorphic, they must possess the same structure, meaning that there's a one-to-one correspondence between the elements of the groups that also preserves the group operation. To prove that these two groups are not isomorphic, one typically needs to: 1. **Compare the Orders of the Groups:** The order of a group is the number of elements in the group. If two groups are isomorphic, they must have the same order. 2. **Analyze the Group Structure:** Determine whether they share properties such as the number of elements of each order. For example: - Calculate $ \mathbb{Z}_{10}^{\times} $. - Calculate $ \mathbb{Z}_{12}^{\times} $. - Compare the orders of the units in each group. - Look for differences in the structure of each group. This problem requires knowledge of group theory, including concepts such as group order, group isomorphism, and units modulo $ n $.$

Transcribed Image Text:### Title: Proving Non-Isomorphism Between $ \mathbb{Z}_{10}^{\times} $ and $ \mathbb{Z}_{12}^{\times} $ **Problem Statement:** Prove that $ \mathbb{Z}_{10}^{\times} $ is not isomorphic to $ \mathbb{Z}_{12}^{\times} $. **Explanation:** In this problem, we are tasked with proving that the group of units modulo 10, denoted as $ \mathbb{Z}_{10}^{\times} $, is not isomorphic to the group of units modulo 12, denoted as $ \mathbb{Z}_{12}^{\times} $. Here $ \mathbb{Z}_n^{\times} $ stands for the multiplicative group of integers modulo $ n $. An *isomorphism* between two groups is a bijective (one-to-one and onto) group homomorphism. For two groups to be isomorphic, they must possess the same structure, meaning that there's a one-to-one correspondence between the elements of the groups that also preserves the group operation. To prove that these two groups are not isomorphic, one typically needs to: 1. **Compare the Orders of the Groups:** The order of a group is the number of elements in the group. If two groups are isomorphic, they must have the same order. 2. **Analyze the Group Structure:** Determine whether they share properties such as the number of elements of each order. For example: - Calculate $ \mathbb{Z}_{10}^{\times} $. - Calculate $ \mathbb{Z}_{12}^{\times} $. - Compare the orders of the units in each group. - Look for differences in the structure of each group. This problem requires knowledge of group theory, including concepts such as group order, group isomorphism, and units modulo $ n $.

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