Find the order of each of the following elements. [from #2, 4.5] a) 5 € Z12 b) -i E C*
Find the order of each of the following elements. [from #2, 4.5] a) 5 € Z12 b) -i E C*
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#5. Thanks.
![### Finding the Order of Elements
#### Problem 5:
Find the order of each of the following elements.
Refer to sections [#2, 4.5] for additional context and examples.
a) \( 5 \in \mathbb{Z}_{12} \)
b) \( -i \in \mathbb{C}^* \)
---
**Explanation:**
This problem requires determining the order of specific elements in given mathematical structures.
1. **For the element \( 5 \in \mathbb{Z}_{12} \):**
- \(\mathbb{Z}_{12}\) denotes the cyclic group of integers modulo 12.
- The order of an element \(a\) in a group is the smallest positive integer \(k\) such that \(a^k = e\) (where \(e\) is the identity element).
2. **For the element \( -i \in \mathbb{C}^* \):**
- \(\mathbb{C}^*\) denotes the multiplicative group of non-zero complex numbers.
- The order of an element \(g\) in this group is the smallest positive integer \(m\) such that \(g^m = 1\) (where 1 is the identity element under multiplication).
To solve these problems, you would usually:
- Calculate or find the smallest exponent that satisfies the given conditions for each element in their respective group.
Refer to your course notes or textbook for methodologies specific to cyclic groups and multiplicative groups of complex numbers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F839fadf6-5483-4d36-a48d-0531b5d15d41%2Fshu135_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Order of Elements
#### Problem 5:
Find the order of each of the following elements.
Refer to sections [#2, 4.5] for additional context and examples.
a) \( 5 \in \mathbb{Z}_{12} \)
b) \( -i \in \mathbb{C}^* \)
---
**Explanation:**
This problem requires determining the order of specific elements in given mathematical structures.
1. **For the element \( 5 \in \mathbb{Z}_{12} \):**
- \(\mathbb{Z}_{12}\) denotes the cyclic group of integers modulo 12.
- The order of an element \(a\) in a group is the smallest positive integer \(k\) such that \(a^k = e\) (where \(e\) is the identity element).
2. **For the element \( -i \in \mathbb{C}^* \):**
- \(\mathbb{C}^*\) denotes the multiplicative group of non-zero complex numbers.
- The order of an element \(g\) in this group is the smallest positive integer \(m\) such that \(g^m = 1\) (where 1 is the identity element under multiplication).
To solve these problems, you would usually:
- Calculate or find the smallest exponent that satisfies the given conditions for each element in their respective group.
Refer to your course notes or textbook for methodologies specific to cyclic groups and multiplicative groups of complex numbers.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
