**Title: Isomorphism of Multiplicative and Cartesian Groups****Objective:**Show that the multiplicative group \(\mathbb{Z}_8^\times\) is isomorphic to the group \(\mathbb{Z}_2 \times \mathbb{Z}_2\).**Problem Statement:**The task is to demonstrate that the multiplicative group of units modulo 8, denoted \(\mathbb{Z}_8^\times\), is structurally identical to the direct product of the cyclic group of order 2, denoted \(\mathbb{Z}_2\), with itself.This means we need to find a bijective homomorphism:\[ f: \mathbb{Z}_8^\times \to \mathbb{Z}_2 \times \mathbb{Z}_2 \]which is a function that preserves the group operation, ensuring that the structure of \(\mathbb{Z}_8^\times\) matches that of \(\mathbb{Z}_2 \times \mathbb{Z}_2\).To begin, let's examine the elements of each group.### Elements:- \(\mathbb{Z}_8^\times\): The group of units modulo 8 consists of those integers less than 8 that are coprime to 8. These elements are {1, 3, 5, 7}.- \(\mathbb{Z}_2\): The cyclic group of order 2 consists of the elements {0, 1}.- \(\mathbb{Z}_2 \times \mathbb{Z}_2\): This direct product consists of the ordered pairs {(0,0), (0,1), (1,0), (1,1)}.### Verification:Now, we will establish the isomorphism by verifying the group structure.1. **Group Operation Preservation:** - For \(\mathbb{Z}_8^\times\), the operation is multiplication modulo 8. - For \(\mathbb{Z}_2 \times \mathbb{Z}_2\), the operation is component-wise addition modulo 2.2. **Bijection and Homomorphism:** - Define a mapping \( f \) from \(\mathbb{Z}_8^\times\) to \(\mathbb{Z}_2 \times \mathbb{Z}_2\).

We know that if two groups are isomorphic than they have same number of elements i.e. their…

Answered: Show that the multiplicative group Z is…

Math

Advanced Math

Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

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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