Find all non-trivial direct factorizations of (1) the four-group (V={e, (12)(34), (13)(24), (14)(23)}) and (2) the direct product of two cyclic groups of order 3 (3 3) .

Q: Let G be cyclic group of order 5p where p is a prime with p not equal to 5 . Prove that the number…

A: Answer for (a)

Q: Prove that a group of order 7is cyclic.

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Q: A B D E (iii), (iv) (ii) (i) All None Which of the following are true? (i): If (R,+) is an abelian…

A: (i) is true as it forms a trivial ring. (ii) False, as they are isomorphic as groups under + but not…

Q: (7) Chapter 7, exercise 5.4: Describe the centralizer Z(o) of the permutation o (153) (246) in the…

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Q: Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List…

A: Solution given below

Q: Chapter 6, exercise 4.1: Write the product x²yx-¹y-¹x³y³ in the form x¹y¹ in the dihedral group Dn.

A: Solution:    Given equation is x2yx-1y-1x3y3,    and we have to find products in the form of xiyj in…

Q: (1) Compute the fundamental group of X = R³\{x – axis}U {y – axis} U{z – axis}.

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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: QUESTION 11 Let G= be a cyclic group of order 8. Find all the distinct subgroups of G and list the…

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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: (a) Let G = {e, a, . .. , aº | a10 = e} be the cyclic group of order 10. For which m, is G = (am)?…

A: As per our guideline we are supposed to answer only first asked  question. Kindly repost other…

Q: (b) |G| = 27 = 33 and |G'| = 125 = 53, but neither G nor G' are cyclic, show that the order of any…

A: Given that : G = 27 = 33 and G' = 125 = 53 To prove : G⊕G' < 226 In-order to value these…

Q: QUESTION 6 Prove that a cyclic group G= is Abelian.

A: Given cyclic group G = <a>

Q: 1. Z20 2. U(11)

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Q: For each of the following groups, find the order of the group and the order of each element in the…

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Q: List all elements of the group U(15). Is this group cyclic?

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Q: Use Lagrange's Theorem to show that if G has prime order, then G is cyclic. Even better, show that…

A: Given a group G with a finite order, the order (or size) of any subgroup H of G divides the order of…

Q: 4.2. An nth root of unity is a complex number z such that z" = 1. (a) Prove that the nth roots of…

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Q: UESTION5 Show that U(9)is cyclic and find all the generators of U(9).

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Q: Show that x is not a generator of the cyclic group (Z3[x]/<x3 + 2x + 2>). Find one such…

A: The given cyclic group is: Z3x/x3+2x+2*. We have to show that x is not the generator of the given…

Q: (1) Let G be a group of order 94 = 2 x 47. Let Z(G), as usual, denote the center of G. (a) For each…

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Q: (a) Prove that a*b=a+b-a-b is an operation on the set S = R\{1}. (Show that, if a*b = 1, then one of…

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Q: State Fermat's Little Theorem. In the groun (7/1017)x find 5100 5101 and 5202 (101 is a prime…

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