construct cayley table (Z(9) , circle times )and verify its an abelian group. **Transcription for Educational Website:**---**Constructing the Cayley Table for $ (\mathbb{Z}_9, \circ) $**This task involves the construction of a Cayley table for the set $ \mathbb{Z}_9 $ under a given operation $ \circ $. We aim to verify if this set forms an Abelian group.**Steps:**1. **Define the Set**: $\mathbb{Z}_9$ represents the set of integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ with operations performed modulo 9.2. **Operation $ \circ $**: Clearly define the operation $ \circ $ (commonly addition or multiplication modulo 9).3. **Construct the Cayley Table**: - Create a table with rows and columns labeled by the elements of $\mathbb{Z}_9$. - Fill in each cell with the result of the operation $ \circ $ applied to the row and column headers.4. **Verify Abelian Properties**: - **Closure**: Ensure every operation result is within $\mathbb{Z}_9$. - **Associativity**: Confirm that $(a \circ b) \circ c = a \circ (b \circ c)$ for all $a, b, c$ in $\mathbb{Z}_9$. - **Identity Element**: Identify an element $e$ such that $e \circ a = a \circ e = a$ for all $a$. - **Inverse Element**: For each element $a$, find an element $b$ such that $a \circ b = b \circ a = e$. - **Commutativity**: Verify $a \circ b = b \circ a$ for all $a, b$ in $\mathbb{Z}_9$.By following these steps, one can demonstrate that $ (\mathbb{Z}_9, \circ) $ is an Abelian group if all properties are satisfied.--- No graphs or diagrams are included.

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Answered: Construct the Cayley table for (Zo)…

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Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

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

construct cayley table (Z₍₉₎ , circle times )and verify its an abelian group.

**Transcription for Educational Website:**

---

**Constructing the Cayley Table for $ (\mathbb{Z}_9, \circ) $**

This task involves the construction of a Cayley table for the set $ \mathbb{Z}_9 $ under a given operation $ \circ $. We aim to verify if this set forms an Abelian group.

**Steps:**
1. **Define the Set**: $\mathbb{Z}_9$ represents the set of integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ with operations performed modulo 9.

2. **Operation $ \circ $**: Clearly define the operation $ \circ $ (commonly addition or multiplication modulo 9).

3. **Construct the Cayley Table**:
- Create a table with rows and columns labeled by the elements of $\mathbb{Z}_9$.
- Fill in each cell with the result of the operation $ \circ $ applied to the row and column headers.

4. **Verify Abelian Properties**:
- **Closure**: Ensure every operation result is within $\mathbb{Z}_9$.
- **Associativity**: Confirm that $(a \circ b) \circ c = a \circ (b \circ c)$ for all $a, b, c$ in $\mathbb{Z}_9$.
- **Identity Element**: Identify an element $e$ such that $e \circ a = a \circ e = a$ for all $a$.
- **Inverse Element**: For each element $a$, find an element $b$ such that $a \circ b = b \circ a = e$.
- **Commutativity**: Verify $a \circ b = b \circ a$ for all $a, b$ in $\mathbb{Z}_9$.

By following these steps, one can demonstrate that $ (\mathbb{Z}_9, \circ) $ is an Abelian group if all properties are satisfied.

---

No graphs or diagrams are included.

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