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)
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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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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