m and n, mn, such that: and U(m) ~U(n). Find integers a) |U(m)| = |U(n) b) |U (m)| = |U(n) and U(m) U(n).

Q: QUESTION 4 a) Find the order of each element in Z15/(5). b) Find n such the this group is isomorphic…

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Q: Let G = . The smallest subgroup of G containing a^8 and a^12 is generated by * a^12 a^4 a^6 a^2…

A: Smallest subgroup a^m & a^n  is a^gcd(m,n) = a^gcd(8,12) = a^4 So option 2nd are correct

Q: Show that U(8) and U(10) are not isomorphic.

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Q: 27. Payton ran l miles on Monday, 2 miles on 7. miles on Wednesday. What is the 8. Tuesday, and 1-…

A: topic -median of distance median is the value in the middle when data is arranged in ascending or…

Q: QUESTION 12 Given a = (1 3 5 6 7)(2 3 5 6 7) in Sg. a) (3) Write a as a cycie or a product of…

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Q: Draw the subgroup lattice for Z28-

A: Draw the subgroup lattice for Z28

Q: QUESTION 7 Show that the special linear group, SL(2, R) is non -Abelian.

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Q: QUESTION 3 Find the oder of the indicated element in indicated quotient group. a) 2 + (8)in Z12/ (8)…

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Q: Let G = Z6 and H = (3). Write out he group table for the quotient group G/H.

A: Let G = Z6 and H = <3>. Write out he group table for the quotient group G/H.

Q: QUESTION 9 Draw the subgroup lattice diagram for Z60

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Q: Find all possible homomorphisms form Z10 → Z4. Then find the kernel of each of these homomorphisms.

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Q: QUESTION 10 Use LaGrange's Theorem to prove that a group G of order 11 is cyclic.

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Q: (7) The dihedral group is the subgroup D4 of S4 of order 8 consisting of permutations induced by…

A: Solution 7: By Lagrange's Theorem, the possible orders are 1,2,4 and 8 The only subgroup of order…

Q: Consider the group G = {x E R|x # -1/2} under the binary operation*: x * y = 2xy – x +y– 1. An…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Find all the generators of Z24-

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Q: QUESTION 15 a) How many elements of order 2 are there in S5? b) How many elements of order 2 are…

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Q: QUESTION 9 Find up to isomorphism all Abelian groups of order 36.

A: We have to find the all abelian groups of order 36 up to isomorphism.

Q: Let G be a group. Let x,y e G be such that O(x) = 4, O(y) = 2, x^3 y = yx. Then O(xy) is 6. Infinity…

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Q: QUESTION 5 a) Show that S5 is a non-Abelian group. b) Give an example of a non-trivial Abelian…

A: (a) To show that S5 is non abelian group.

Q: Find the inverse of each element in Z7.

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Q: Which of the following is cyclic? * O U30 O U50 O U20 O U40

A: B is the option:-U(50) As per the guidelines we can only solve one question at a time. Thank you.

Q: Q Let > be a o- Algebra on X. Assume that y,µ:) → [0, 0) are two measures on X.Then a. 3y – 2µ is…

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Q: (3) Chapter 2, exercise 8.5: A finite group contains an element x of order 10 and also an element Y…

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Q: If A3x4 with Nullity(A) = 1, then Rank (AT) = O 1 2

A: Given matrix is A3x4 with nullity(A) = 1

Q: Let A = {1,2,3} and B = {x,y}. %3D %3D (a) How many functions are there from A to B? (b) How many…

A: Given that                 A=1, 2, 3   and B=x, y We have to find in how many functions are there…

Q: Use the left regular representation of the quaternion group Q8 to produce two elements of Sg which…

A: Fix the labelling of Q8 , Take elements 1, 2, 3, 4,  5, 6, 7, 8 are 1, -1, i, -i, j, -j, k, -k…

Q: QUESTION 6 Show that U(14) and U(18) are isomorphic..

A: We have to show that the group U(14) and U(18) are isomorphic.

Q: x Which of the following statements is/are true about the graph of R(x) = x² 4 - A. As the value of…

A: Introduction: For a rational function, vertical asymptote is quite natural. The points where the…

Q: QUESTION 8 Find the inverse of each element in D3.

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Q: Find the square of each element in D4-

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Q: QUESTION 6 Let G = {1,-1,i.-i}. Given that G with the operation of multiplication of complex numbers…

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Q: Let GL(2,R) be the general linear group of 2 by 2 matrices. 1 x is a) Show that o: R GL (2,R) where…

A: We first prove this is a homomorphism.

Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: 4 In the group GL(2, Z¡), inverse of A = 3) This option

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Q: QUESTION 4 Find all the distinct left cosets of H={(1), (1 2)(3 4)} in D4-

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Q: QUESTION 14 Find up to isomorphism all Abelian groups of order 18.

A: To find- Find up to isomorphism all Abelian groups of order 18.

Q: QUESTION 10 Determine the order of each element in U(10).

A: Given: A group U(10)=1, 3, 7, 9. To determine: The order of each element in U(10).