2. Let (G,) be a group of order n, that is, |G|=n. Suppose that a, b E G are given. Find how many solutions the following equations have (your answer may depend on n) in G. a X.ba X².b = (X is the variable) X. a = b.Y (X,Y are the variables) i) ii)

Q: HWGrouPA Prove the fallouwing identity. secA.SecB.CSCA.CSCB SectAtB) CSCA.CSCB-SecA. secB

A: sec(A+B)

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: Use models to help you divide. 1. 4)76 2. 2)94 4. 7)238 5. 6)426 . Ifn 3 = 57, what is the value of…

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Q: 07 : Pove equeivo lance. Partition equeivalonce Set casses the xelation. of the that from on 9. sef.

A:

Q: 5. Let B = {-1, -i, i, 1}; where i² = multiplication. -1. Show (B,*) is a group where * represents 4

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Q: 3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only ir gcd(a/d, b/d) =…

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Q: (a). H, K≤ G and HGH* K≤G. (b). HG and KAG⇒ H✩ KAG.

A:

Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: also compute η2 for a. and r2 for b.

A: Eta square measure the effect size of the observations. It is calculated by the combined value of…

Q: If R = {(r, y) E R × R: r- y =0}. Then list of all members of [4] is: Select one: O a. none of these…

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Q: Let 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 and B= 2 3 4 5 1 7 8 6 13 8 7 6 5 2 4 Write a, B, and aß as a.…

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Q: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

Q: 7.) Name 3 different (single) transformations that would map this figure to itself. -2 3 A. B. -2-…

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Q: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

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Q: a. Find C if C = AʘB.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Melissa claims the triangles are similar. Is her claim correct? If her claim is correct, state the…

A: Yes Melissa's claim is correct.

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: It is given that x and y be two vertices of a Cayley digraph. We can write two paths as a1,a2,a3,…an…

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

A:

Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

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Q: 7. Prove that the graphs G & H are isomorphic. 1 3 a 5 2 H

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Q: L[(2t – e^5t)^3]

A: Please see the next step

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} where a = (1 2 3 4) and b =…

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Q: U(11) is cyclic generated by  Select one: a. 2 b. 10 c. 9 d. 5

A: Correct option a. 2

Q: 13. Find groups that contain elements a and b such that la| = |b| = 2 and a. labl = 3, b. labl = 4,…

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

A: Answer : 1 right cancelation law

Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

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Q: 2 3 4 1 4 3 2 3 4 1) 1 2 2. Let X = {1,2, 3, 4}. Consider the following elements of P(X): a = (; 2 3…

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Q: Construct the group table for U (24). Find the order of each element in U(24).

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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

A:

Q: Discuss the mapping properties of for ny 2 -15 N

A: The mapping properties of z1n has to be discussed. Here the range is n≥2.

Q: y P(14r-8) (12y)

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Q: 4. Let G be a group and g E G be an element of order 8. Find the order of the following elements.…

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Q: 34. Suppose a and b are group elements such that |a| = 2, b ‡ e, and aba = b². Determine [b].

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Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________

Q: Please do part C and D and please show step by step and explain

A: (c) Suppose the given structure {G = {a, b, c, d}, ∘} forms a group.       Now, a ∘ a = a. . . . (1)…