e 2πik/n (1 ≤ k ≤ n) is a group under multiplication.

Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: To Prove: There is no simple group of order 525=3·52·7.

Q: for each of the group Z2,Z4,Z6 INDICATE WHICH CYCLIC. FOR those cyclic list  all the generators?

A:

Q: Suppose F1 and F2 are distinct reflections in a dihedral group Dn.Prove that F1F2 ≠ R0.

A: Given: F1 and F2 are distinct reflections in a dihedral group Dn. To Prove: F1F2≠R0.

Q: 5-) Find all elements in his which has multiplcn the interse, then find their inverses

A: The objective of the question is determine the elements of Z15  and evaluate the   multiplicative…

Q: Give all the possible elementary divisors of a group of order 40.

A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5

Q: Assume that (G, ) is a group, that geG, and that t is the smallest positive integer such that g' =…

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Q: Let a EC be a primitive nth root of unity.

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Q: (a) Explain why it is impossible for any set of (real or complex) numbers which contains both 0 and…

A: To solve the given problem, we use the defination of group.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

A: We have to show that the permutation of m letters, m≥3 that is Sm is not abelian. For proving the…

Q: Show that the symmetry group in R3 of a box of dimensions 2n x3n x 4nis isomorphic to Z2 ⊕Z2 ⊕ Z2.

A: In this solution, we will determine the symmetry group of a box with dimensions 2ⁿ × 3ⁿ × 4ⁿ. The…

Q: If G = is a distant group and f: G → G is an automorph, show that f(a) is a generator of G, i.e. G =

A: We need to show every element of G is some power of f(a)

Q: Exercise 7.19. Let G be a finite group, H ≤ G, |H| = 2k + 1 for some non-negative integers k, and…

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Q: Q4: prove or disprove each of the following a) Every group has one generator. b) Every group of…

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Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

A:

Q: 3. Let C be the code consisting of solutions to AxT = 0, where %3D 1 10 1 0 1 1 1 0 0 0 1 1 1 1 A =…

A: (i) Consider the given matrix, A=111010111000111 Substitute the value int the equation AxT=0, so we…

Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

A:

Q: Prove that there is no simple group of order 315 = 32 .5 . 7.

A: It has to be proved that there is no simple group of order 315=32·5·7.

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that  G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: Let G= { a+bie C: a and bare rational numbers, not both zero}. Prove that G with the operation of…

A:

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: G is the group of nth roots of unity under complex multiplication Zn is the group of integers…

A: Consider the given information.

Q: 1. Show that every group of prime order is simple.

A:

Q: Prove roots of unity are a subgroup

A:

Q: Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of…

A: Given, The set of 4 elements are 1,-1,i,-i.It is to be shown that this set is group.

Q: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: Prove that each element of the alternating group An can be written as the product of a finite number…

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Q: For each of the following values of ∆, find all elements of the ideal class group of G=C_(delta)…

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Q: Prove Z(60) is cyclic

A: To show that Z(60) is cyclic. It is enough to show that there is a generator which generates Z(60).…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: 14. Prove that the set of all rational number of the form 3"6" | m,nEZ} js a group under…

A: Denote with

Q: Abstract Algebra: Prove that Aut(Z/18Z) is cyclic.

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Q: Let H be the subset of C consisting of the nth roots of unity. Prove that H is a group under…

A: