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4. Prove or disprove the following statement. U(8) is cyclic. [from #1, 4.5]

4. Prove or disprove the following statement. U(8) is cyclic. [from #1, 4.5]

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 33EQ
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### Problem Statement 4. Prove or disprove the following statement: \( U(8) \) is cyclic. \([from #1, 4.5]\) --- #### Explanation: The notation \( U(8) \) refers to the group of units modulo 8, which consists of integers less than 8 that are relatively prime to 8, under multiplication modulo 8. To determine if \( U(8) \) is cyclic, one would need to find if there exists an element in \( U(8) \) that can generate all other elements of the group through its powers. The typical steps to prove or disprove this would include: 1. **Identify the Elements of \( U(8) \):** List out all integers less than 8 that are coprime to 8. 2. **Check for a Generator**: Determine if any of these elements can generate all other elements of the group under multiplication modulo 8. Graphical representation or detailed steps might be provided in other sections or diagrams to illustrate the process of checking for generators.
Transcribed Image Text:### Problem Statement 4. Prove or disprove the following statement: \( U(8) \) is cyclic. \([from #1, 4.5]\) --- #### Explanation: The notation \( U(8) \) refers to the group of units modulo 8, which consists of integers less than 8 that are relatively prime to 8, under multiplication modulo 8. To determine if \( U(8) \) is cyclic, one would need to find if there exists an element in \( U(8) \) that can generate all other elements of the group through its powers. The typical steps to prove or disprove this would include: 1. **Identify the Elements of \( U(8) \):** List out all integers less than 8 that are coprime to 8. 2. **Check for a Generator**: Determine if any of these elements can generate all other elements of the group under multiplication modulo 8. Graphical representation or detailed steps might be provided in other sections or diagrams to illustrate the process of checking for generators.
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