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2.4 Prove theorems 1. Identity elements of a group is unique. 2. Every element of a group has unique inverse element. 3. Assume there is binary operation o that is defined for a set T which consist of two elements. This operation has associative property. Does this operation o hold commutativity?

2.4 Prove theorems 1. Identity elements of a group is unique. 2. Every element of a group has unique inverse element. 3. Assume there is binary operation o that is defined for a set T which consist of two elements. This operation has associative property. Does this operation o hold commutativity?

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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2.2 Cayley table / Composition table 1. Build composition table for a group (< i > = {i,-1, -i,1}, x). i - is imaginary unit. 2. Build composition table for a symmetric group S3. 3. Build composition table for a group (Z*15, × mod 15) (Hint: Z*15 - is the set of elements with multiplicative inverses mod 15 (set of units in Z15)) 4. Build composition table for a dihedral group D4 (Hint: You might need read additional materials to solve this.) 2.3 Cyclic group, order of a group/element 1. Write a definition of a cyclic group. Provide two examples of cyclic group. 2. Calculate the order of the symmetric group Ss 3. Calculate the order of the group (Z7, +) 4. Calculate the order of the element 4 in group (Z*18, × mod 18) 5. Calculate the order of element 23 in a group (Z:1, +) 6. Calculate the order of the element t12 in group (S3, 0) 2.4 Prove theorems 1. Identity elements of a group is unique. 2. Every element of a group has unique inverse element. 3. Assume there is binary operation o that is defined for a set T which consist of two elements. This operation has associative property. Does this operation o hold commutativity?
Transcribed Image Text:2.2 Cayley table / Composition table 1. Build composition table for a group (< i > = {i,-1, -i,1}, x). i - is imaginary unit. 2. Build composition table for a symmetric group S3. 3. Build composition table for a group (Z*15, × mod 15) (Hint: Z*15 - is the set of elements with multiplicative inverses mod 15 (set of units in Z15)) 4. Build composition table for a dihedral group D4 (Hint: You might need read additional materials to solve this.) 2.3 Cyclic group, order of a group/element 1. Write a definition of a cyclic group. Provide two examples of cyclic group. 2. Calculate the order of the symmetric group Ss 3. Calculate the order of the group (Z7, +) 4. Calculate the order of the element 4 in group (Z*18, × mod 18) 5. Calculate the order of element 23 in a group (Z:1, +) 6. Calculate the order of the element t12 in group (S3, 0) 2.4 Prove theorems 1. Identity elements of a group is unique. 2. Every element of a group has unique inverse element. 3. Assume there is binary operation o that is defined for a set T which consist of two elements. This operation has associative property. Does this operation o hold commutativity?
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