Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic groups of orders n₁, ₂, ..., n where n₁+1 divides n, for i for i = 1, 2, ..., t - 1. (This exercise is re- ferred to in this chapter.)

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: Describe the following points on the complex plane: (a) Re(iz +3 – i2) = 1, (b) |z – i| = |z+ 1|,

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Q: Sylow Theorems

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Q: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: The characteristic polynomial of the following matrix have been provided. Find the Jordan canonical…

A: Given: charAt=1-t21+t2,  A=-2-1001000-201001-51 To find the Jordan canonical form of A.

Q: 3.4. Let G, H be groups, and assume that G H x G. Can you conclude that H is trivial? (Hint: No. Can…

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Q: 4. Let p be a prime. Find all abelian groups of order på that contain no elements of order p up to…

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Q: 9. Determine with proof whether the following statements are true or false. (a) There exists an…

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Q: Express the following as products of cyclic gr a) Z140 ») U (140) =) Z400 H) U (400).

A: To express the given groups as products of cyclic groups, we need to find the prime factorization of…

Q: 3. Infinite, abelian (and additive), and so that for every element a, a+a=0. (Hint: Think of…

A: Note the question asks us for an example of an infinite group such that sum of two elements is…

Q: 7. Find order of all the elements of the Group = {1,–1, i, -i}. The binary operator * is defined as…

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Q: True or False: (a) 3Z = 9Z (b)Let p be a prime number. Then Zp × Zp = Z,² (c) Every subgroup of a…

A: We have to state whether the following statements are true or false. We have  and . We have to show…

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

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: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

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Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: . Find all of the abelian groups a) of order 200 up to isomorphism b) of order 720 up to…

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Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: What are all abelian groups of order 81 up to isomorphism? Are there any Theorems I can use to…

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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: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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