6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10×Z85 c) Z4xZ25x Z, d) ZnxZ2 x Z65 -21

Q: 54 The groups An and Sn are not solvable for all n ≥ 5.

A: Solution:-

Q: Q/ How many non-trivial subgroups in s, ? a) 2 b) 3 c) 4

A: S3 is the set of permutations on the set {1, 2, 3}S3={e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) }The…

Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

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Q: · In a group, prove that (ab) = b-'a-|

A: As you asking for question number 7 , I solve for you.

Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: B. Write at least five elements of each of the following cyclic groups. 1. 25Z under multiplication…

A: 25Z under multiplication: {-50, -25, 25, 50, 75}

Q: G is Klein's 4-group%3D{e,a,b,c}, H=, then [G:H] is 3 4 O 1 O 2 None of them

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Q: 11. Prove that every Cayley table is a Latin square for a group. That is, each element of the group…

A: To prove, each element of the group appears exactly once in each row and each column of a Cayley…

Q: In the group Zg compute, (a) 6 + 7, and (b) 2-'.

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Q: 10. Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not…

A: Given that,   (a) Group of order 40. Let G=40,  i.e, G=40=23.5, Since, 5-Sylow subgroup, i.e…

Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: In a group G,let a,b and ab have order 2.show that ab=ba

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Q: Give two reasons why the set of odd permutations in Sn is not a subgroup.

A: To show that, The set of odd permutations in Sn is not a subgroup. A subset H of the group G is a…

Q: What is the order of the element $(\overline{2}, \overline{9})$ in $Z_{4} \times U_{10}$ is (…

A: The set ℤ4 ,i.e the set of congruence modulo 4 is an additive group. Add the element 2 with itself…

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * e b с a e e а…

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Q: 10) Which of the following is a group? * O (Z,*), a* b = a + b - 1 va, b e Z O (Z,*), a* b = a - 2b…

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Q: 3. Show that the following permutations: e = 12345 67 8 a = 12345 678, B = 1 2 3 4 5 678 Y =…

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Q: Show, by example, that in a factor group G/H it can happen thataH = bH but |a| ≠ |b|.

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Q: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

A: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

Q: A. B. mLA = mLD. (Hint: Use the result in Example 1.) 11. A circle passes through the vertices of…

A: Given that a circle passes through the vertices of ABCD and AB = CD. Here we have to prove m<A =…

Q: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?

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Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140

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Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

Q: In a group (G, 0) the elements a and b commute. Also O (a) = m and O (b) = n, are prime to each %3D…

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Q: In the group Z8 compute, (a) 6+7, and (b) 2-1.

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Q: 16. A class consist of 15 men and 20 women, in how many ways can a group of 5 be formed if it has to…

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Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

Q: In the group U (16) compute, (a) 5-7, and (b) 3-1

A: M(16) refers to th e multiplication group of non-negative integers (mod 16) that are less than 16…

Q: In a group G, the law ab-Dac implies b-c is called..

A: Answer : 1 right cancelation law

Q: How would we show that H is cyclic?

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Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10x Zgs c) Z4xZ2sx Z, d) Z22XZ21 xZ6s

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Q: 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are…

A: Given:  Let G be a group. ZG be its center.  We know that ZG=z∈G: ∀g∈G,zg=gz ....i First we will…

Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…

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Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups  are cyclic groups. A group G is cyclic if G=<g> for some g in G, where…

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

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Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: In the group U (I6) compute, (a) 5.7, and (b) 3

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Q: In a group (G, 0) the elements a and b commute. Also O (a) = m and O (b) = n, are prime to each…

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Q: 2. 20 Wnat is the loargest order of an element in Zs Ze O Z 2) Exhibit an element with this arder,

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Q: (b) How many elements of the permutation group Se map 2 to 2 and 5 to 5, while the re- maining…

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Q: Which of the following groups are cyclic? Justify. (a) G = U(10) = {k e Z10 : ged(k, 10) = 1} =…

A: We know that  1)Every cyclic group is almost countable  2) Every finite cyclic group is isomorphic…

Q: Express the permutation (2 4 5)(1 3 5 4)(1 2 5) as a single cycle or a s a product of cycles and how…

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Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}

A: We have given a group Z4 ,  and we have to find which of the following sets {0,2},{0,3},{0,1} is a…

Q: In Z24 the number of all subgroups is 8 O 6.

A: We are asked to find the subgroup of z24