**Problem 5: Finding Right Cosets**Find the right cosets of the subgroup $ H $ in $ G $ for $ H = \{(0,0), (1,0), (2,0)\} $ in $ \mathbb{Z}_3 \times \mathbb{Z}_2 $.---In this problem, we are asked to identify the right cosets of a given subgroup $ H $ within the group $ \mathbb{Z}_3 \times \mathbb{Z}_2 $. The subgroup $ H $ consists of the elements $(0,0)$, $(1,0)$, and $(2,0)$.**Explanation of Symbols:**- $ \mathbb{Z}_3 $ denotes the cyclic group of integers modulo 3. Its elements are $\{0, 1, 2\}$.- $ \mathbb{Z}_2 $ denotes the cyclic group of integers modulo 2. Its elements are $\{0, 1\}$.- $ \mathbb{Z}_3 \times \mathbb{Z}_2 $ is the direct product of these two groups, forming a set of ordered pairs where the first element is from $\mathbb{Z}_3$ and the second is from $\mathbb{Z}_2$.**Objective:**To find all the right cosets of $ H $, calculate $ xH $ for each element $ x $ in $ G = \mathbb{Z}_3 \times \mathbb{Z}_2 $.---

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Answered: 5. Find the right cosets of the…

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5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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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Question

**Problem 5: Finding Right Cosets**

Find the right cosets of the subgroup $ H $ in $ G $ for $ H = \{(0,0), (1,0), (2,0)\} $ in $ \mathbb{Z}_3 \times \mathbb{Z}_2 $.

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In this problem, we are asked to identify the right cosets of a given subgroup $ H $ within the group $ \mathbb{Z}_3 \times \mathbb{Z}_2 $. The subgroup $ H $ consists of the elements $(0,0)$, $(1,0)$, and $(2,0)$.

**Explanation of Symbols:**

- $ \mathbb{Z}_3 $ denotes the cyclic group of integers modulo 3. Its elements are $\{0, 1, 2\}$.

- $ \mathbb{Z}_2 $ denotes the cyclic group of integers modulo 2. Its elements are $\{0, 1\}$.

- $ \mathbb{Z}_3 \times \mathbb{Z}_2 $ is the direct product of these two groups, forming a set of ordered pairs where the first element is from $\mathbb{Z}_3$ and the second is from $\mathbb{Z}_2$.

**Objective:**

To find all the right cosets of $ H $, calculate $ xH $ for each element $ x $ in $ G = \mathbb{Z}_3 \times \mathbb{Z}_2 $.

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