If H and K are subgroups of G, IH|= 16 and |K|=28 then a possible value of |HNK| is * 6 4 O 16
Q: Show that if H and K are subgroups of G then so is H ∩ K.
A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…
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A: This is a good exercise in working with cosets. We first find out the subgroup $H$ and then working…
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A: To find All subgroups of z2*z4
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Q: Q2.3 Question 1c Let G = Są and let H = {o € S4 | o (2) = 2}. Then %3D O H is not a subgroup in G O…
A: Solution.
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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.
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Q: W6 Assume that H, k, and k are SubgrouPs of the group G and k, , Ka 4 G. if HA k, = HN k Prove that…
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Q: If H and K are subgroups of G, |H|= 18 and |Kl=30 then a possible value of |HNK| is O18 8. O 4
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Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is * O 6 16 8. 4
A: Order of an subgroup should divide order of an group. Intersection of two subgroups again a…
Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is * 18 8 6. 4
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Q: Q7. Suppose that the index of the subgroup H in G is two. If a and b are not in H, then ab ∈ H.…
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Q: (c) Find all subgroups of (Z/2)*3 = Z/2 × Z/2 × Z/2.
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Q: If H and K are subgroups of G, |H]= 18 and |K|=30 then a possible value of |HNK| is * O 8 6. 4 O 18
A: For complete solution kindly see the below steps.
Q: (a) Draw the lattice of subgroups of Z/6Z. (b) Repeat the above for the group S3.
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Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…
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Q: If H and K are subgroups of G, |H|= 16 and IK|=28 then a possible value of |HNK| is * O 16 4 8.
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Q: roblem 9.6 Let G = Z/100 and assume that H C G is a subgroup. xplain why it is impossible that |H|=…
A: Order of a subgroup divides the order of a group
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Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is 8 O 16 4 6
A: Answer is 4.
Q: What is [Z12: (4)]? Find all cosets of the subgroup (4) of Z12. A. {(4), 1 + (4)} B. {(4), 1+ (4), 2…
A: To find the required cosets and index :-
Q: If H and K are subgroups of G. IH|- 20 and IK-32 then a possible value of HNK| is 16 8.
A: This is a question from Group theory concerning the order of a group. We shall use Lagrange's…
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Q: If H and K are subgroups of G, |H|= 16 and IK|=28 then a possible value of |HNK| is * O 16 6. 4 O O…
A: H and K are subgroups of G H=16 and K=28 we have to find the possible value of H∩K
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Q: If H and K are subgroups of G, |H|= 18 and |K]=30 then a possible value of |HOK| is O 4 O 18 O 8
A: Given that H and K are sub-group of G. |H|= 18 |K|=30 To find…
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Q: QUESTION 10 Show that G ={a +bv3: a,b EQ}is subgroup of R under addition.
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Q: f H and K are subgroups of G, IH|= 20 and K|=32 then a possible value of |HOK[ is * O 16
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Q: 3= {m + nV5| m * EZ}CR. ca> Show that s is a subgroup ob (R, +), (6) Show that ib sl, st ES then…
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Q: If H and K are subgroups of G, |H|= 20 and |K]=32 then a possible value of IHNKI is O 2 16
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Q: How many cyclic subgroups of order 2 in Zg O Z2 4 None of them 2 1 3
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Q: If H and K are subgroups of G, H|= 16 and |K|=28 then a possible value of |HNK| is * 4 О 16 6 00 ООО…
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Q: 1- (Z,+) is a subgroup of (Q,+) 2- (Q,+) is a a subgroup of (R, +) 3- (R,+) is a a subgroup of (C,+)…
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Q: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…
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Q: D. Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that…
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Q: Suppose that f:G →G such that f(x) = axa'. Then f is a group homomorphism if %3| and only if a = e…
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- Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.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.In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.
- For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
- 22. If and are both normal subgroups of , prove that is a normal subgroup of .Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?Exercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.
- 23. Prove that if and are normal subgroups of such that , then for allLet G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 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. 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. 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. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.