Abstract Algebra I QUESTION 5a) Find the order of each element in Da/<(1 3)(2 4)>.b) Is this group isomorphic to Z, for some n? Justifyyour answer.

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Answered: UESTION 5 a) Find the order of each…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.3: Permutation Groups In Science And Art (optional)

Problem 7TFE: The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That...

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Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
9. Find all homomorphic images of the octic group.
Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.
Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.

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UESTION 5 a) Find the order of each element in D4/ <(1 3)(2 4)>. b) Is this group isomorphic to Z, for some n? Justify your answer.