TABLE 5.1The Alternating Group A4 of Even Permutations of {1, 2, 3, 4}(In this table, the permutations of A4 are designated as 1, a2, . . . , &12 and an entry kinside the table represents ag. For example, az ag = a6.)a4a6aga10d12(1) = a1(12)(34) = a2(13)(24) = az(14)(23) = a4(123) = a5(243) = a6123478.9.10111246.2387109.123%3D78.611129.318.6 5 12711108.6.9.121011146.(142) = a7(134) = ag(132) = ¤97 5 81011122376 8 51110129.3856129.111019.11121014257(143)10%3Da1012119.24316.8.(234) = ¤11119.10129.(124) = a1231456.1210114.138S=O9 34126507O214387 651 2 (13)(24) = a3(14)(23) = a4(123) = as(243) = a643216.7.58.6.7.8.%3D(142) = a7(134) = ag(132) = a9(143) = a10(234) = a11126.8.8.6.%3D9 111210%3D101211 9116.1012(124) = a12109.11%3DGiven H = {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.a) Show that His a subgroup of A4-b) Find all the left cosets of H in A4.c) Find all the right cosets of H in A4-d) Does aH =Ha for all a in A4?345

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Answered: Given H = a) Show that His a subgroup…

Given H = a) Show that His a subgroup of A4- b) Find all the left cosets of H in A4. c) Find all the right cosets of H in A4. d) Does aH=Ha for all a in A4? {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}.

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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The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
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In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).
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In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
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