37. Suppose G is the group defined by the following Cayley table. 1 4 6. 8 1 3. 4 8. 8. 7 6. 3 7. 1 2 3 1 8. 7. 6 5 8 1 3 6. 4 1 8 7 7. 8 1 3 4 5 6 8 7 6 4 3 2 1 a. Find the centralizer of each member of G. b. Find Z(G). c. Find the order of each element of G. How are these orders arith- metically related to the order of the group? 4) 658N 2. 6 325 3. 214 234 123 45 67980
37. Suppose G is the group defined by the following Cayley table. 1 4 6. 8 1 3. 4 8. 8. 7 6. 3 7. 1 2 3 1 8. 7. 6 5 8 1 3 6. 4 1 8 7 7. 8 1 3 4 5 6 8 7 6 4 3 2 1 a. Find the centralizer of each member of G. b. Find Z(G). c. Find the order of each element of G. How are these orders arith- metically related to the order of the group? 4) 658N 2. 6 325 3. 214 234 123 45 67980
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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![Below is a transcription and explanation suitable for an educational website:
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**Understanding Group Structure: Cayley Table and Exercises**
**Problem 37:**
Suppose \( G \) is the group defined by the following Cayley table:
\[
\begin{array}{c|cccccccc}
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
2 & 2 & 1 & 4 & 3 & 6 & 5 & 8 & 7 \\
3 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 \\
4 & 4 & 3 & 6 & 5 & 8 & 7 & 2 & 1 \\
5 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 \\
6 & 6 & 5 & 8 & 7 & 2 & 1 & 4 & 3 \\
7 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 \\
8 & 8 & 7 & 2 & 1 & 4 & 3 & 6 & 5 \\
\end{array}
\]
**Questions:**
a. Find the centralizer of each member of \( G \).
b. Find \( Z(G) \), the center of \( G \).
c. Find the order of each element of \( G \). How are these orders arithmetically related to the order of the group?
**Explanation of Table:**
This table represents the binary operation for the group \( G \). Each cell at position \((i, j)\) in the table represents the product of elements \( i \) and \( j \).
**Steps for Solutions:**
- **Centralizer:** The centralizer of an element \( g \) in \( G \) is the set of elements in \( G \) that commute with \( g \). To find it, examine each row and column pair for commutative relationships.
- **Center \( Z(G) \):** The center of the group is the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2F775a89c3-7555-4786-8e2b-5281a138bea3%2Fmqx9udw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is a transcription and explanation suitable for an educational website:
---
**Understanding Group Structure: Cayley Table and Exercises**
**Problem 37:**
Suppose \( G \) is the group defined by the following Cayley table:
\[
\begin{array}{c|cccccccc}
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
2 & 2 & 1 & 4 & 3 & 6 & 5 & 8 & 7 \\
3 & 3 & 4 & 5 & 6 & 7 & 8 & 1 & 2 \\
4 & 4 & 3 & 6 & 5 & 8 & 7 & 2 & 1 \\
5 & 5 & 6 & 7 & 8 & 1 & 2 & 3 & 4 \\
6 & 6 & 5 & 8 & 7 & 2 & 1 & 4 & 3 \\
7 & 7 & 8 & 1 & 2 & 3 & 4 & 5 & 6 \\
8 & 8 & 7 & 2 & 1 & 4 & 3 & 6 & 5 \\
\end{array}
\]
**Questions:**
a. Find the centralizer of each member of \( G \).
b. Find \( Z(G) \), the center of \( G \).
c. Find the order of each element of \( G \). How are these orders arithmetically related to the order of the group?
**Explanation of Table:**
This table represents the binary operation for the group \( G \). Each cell at position \((i, j)\) in the table represents the product of elements \( i \) and \( j \).
**Steps for Solutions:**
- **Centralizer:** The centralizer of an element \( g \) in \( G \) is the set of elements in \( G \) that commute with \( g \). To find it, examine each row and column pair for commutative relationships.
- **Center \( Z(G) \):** The center of the group is the
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