List all cosets of (([1]3, [2]6)) in Z3 × Ze
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### 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 \(([](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd95b1c5e-3673-40a0-a87b-179a3e73672c%2Fb26cb465-a388-4c5e-ad20-fdbcc5ef94c1%2Fpbdt1h_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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