### Problem Statement:List all cosets of \((([1]_3, [2]_6))\) in \(\mathbb{Z}_3 \times \mathbb{Z}_6\).---In this problem from abstract algebra, we are asked to list all the cosets of the given element \((([1]_3, [2]_6))\) in the direct product group \(\mathbb{Z}_3 \times \mathbb{Z}_6\). The notation \(\mathbb{Z}_3\) refers to the cyclic group of integers modulo 3, and \(\mathbb{Z}_6\) refers to the cyclic group of integers modulo 6. In \(\mathbb{Z}_3 \times \mathbb{Z}_6\), elements are ordered pairs where the first component is from \(\mathbb{Z}_3\) and the second component is from \(\mathbb{Z}_6\).Each coset of the element \((([1]_3, [2]_6))\) can be obtained by adding \((([1]_3, [2]_6))\) to every element of \(\mathbb{Z}_3 \times \mathbb{Z}_6\).Here are the steps to determine all the cosets:1. **Identify the elements in \(\mathbb{Z}_3 \times \mathbb{Z}_6\)**. - Elements in \(\mathbb{Z}_3\) are \([0]_3, [1]_3,\) and \([2]_3\). - Elements in \(\mathbb{Z}_6\) are \([0]_6, [1]_6, [2]_6, [3]_6, [4]_6,\) and \([5]_6\). - Thus, there are \(18\) elements in \(\mathbb{Z}_3 \times \mathbb{Z}_6\).2. **Construct cosets**: - Start with the given element: \((([1]_3, [2]_6))\). - Add \((([1]_3, [2]_6))\) to each element of \(\mathbb{Z}_3 \times \mathbb{Z}_6\) element-wise. For example: Adding \(([

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Answered: List all cosets of (([1]3, [2]6)) in Z3…

List all cosets of (([1]3, [2]6)) in Z3 × Ze

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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