(! 2 3 45 6 7 8 (3 8 5 67 4 9 2 1) ) in the sym- Let o denote the permutation metric group S, on the set {1,2,3,4, 5,6, 7, 8, 9). (a) Write o as a product of disjoint cycles. (6) Write a as a product of transpositions. (c) Is o even, odd, neither or both?
Q: Let σ denote the permutation � 1 2 3 4 5 6 7 8 9 3 8 5 6 7 4 9 2 1� in the symmetric group S9 on the…
A: Given a permutaion in the symmetric group S9on the set {1,2,3,4,5,6,7,8,9}1 2 3 4 5 6 7 8 93 8 5 6…
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Q: (ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (5 6), (1 2)(3 4)(5 6)} with the…
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Q: The following is a Cayley table for a group G. The order of 4 is: 2 3 5 2 3 4 5 3 4 1 2 4 2 1 3 2 3…
A: According to our company's guidelines I can only answer first question since you have asked multiple…
Q: The following is a Cayley table for a group G. The order of 4 is: 1 2 3 4 1 3 4 5 4 4 5 2 4 1 2 3 4…
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A: Where we have to find out the order and the inverse of given permutation.
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Q: 5. Find the number of generators of the cyclic group Z15
A: To find the number of generators of the cyclic group ℤ15.
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A: If G be a group and a belongs to G then c(a) = { x : ax = xa , where x belongs in G}.
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Q: group. (b) What do you mean by conjugate elements and classes ? Find the classes of the (Rohilkhand…
A: Solution -
Q: In the group S6, consider the elements 1 2 3 4 5 6 4 3 6 5 1 2 1 2 3 4 56 35 4 2 6 1 a = and B = (a)…
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Q: 2. If a is an involution, then Baß is an involution for any transformation B.
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Q: 5. The permutations a = (124)(35) and b = (12)(3)(4)(5) generate a group (G, *) of order 12. (a)…
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Q: 1 2 3 4 5 6 7 8 9 3 8 5 6 7 49 2 1 Q 22 Let o denote the permutation in the sym- metric group S9 on…
A: Please upload the multiple parts separately.I answered here first 3 sub-parts as per our policy.
Q: Prove that in a group, (ab)^2=a^2b^2 if and only if ab=ba.
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Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?
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Q: (a) Express the permutation (2 4 5)(1 3 5 4)(1 2 5) as a single cycle or as a product of cycles. (b)…
A: NOTE:- Only three subparts can be answered at a time as per the company guidelines. (a) Given is the…
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Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}
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A: The given set is S=σ∈S4:σ4=4 is the subset of S4.
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- 11. Show that is a generating set for the additive abelian group if and only ifExercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19
- Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .
- 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .29. State and prove Theorem for an additive group. Theorem : Generalized Associative Law Let be a positive integer, and let denote elements of a group . For any positive integer such that , .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.
- 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.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.23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .