Which of the following is a group? The set of non-zero real numbers under division. The set {1,4,5,7} under multiplication modulo 9. The set of positive even integers under addition. The set {1,2,3,4} under multiplication modulo 5. IfGis a finite group and some element of G has order equal to the size of G, we can say G is that G Cyclic Abelian Not cyclic Not abelian

Q: 6. Consider the set G = {4, 8, 12, 16}. Show that this set is a group under multiplication modulo 20…

A: G = {{4,8,12, 16} Under multiplication module 20 Caley Table 20 4 8 12 16 4 15…

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Q: abouis 13. Let (G, *) be cyclic group of finite order n and let a € G. Prove that ak is a generator…

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Q: O If a group G acts on a set S, every element of S is fixed by the identity of G. O Every group of…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

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Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…

A: Q1.  Third option is correct. Q2. Second option is correct.

Q: Which of the follewing is a group? The set of non-zero real numbers under division. The set…

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Q: For any elements, a and b in a group G, a^(5).b = ba and a^(4) = e, then G is an abelian group…

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Q: We will now consider three irreducible representations for the permutation group P(3): E A B I : (1)…

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Q: 16. True or False: (a). Every element in a cyclic group (G) is a generator of G. (b). A group (G, *)…

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Q: show that if a and b are two different generators of G,* a cyclic group, and therefore abelian, then…

A: If G is cyclic group then order of it's generator is equal to order of group. So, if G has two…

Q: Let G be a finite group with no element of order 3, and let a E G. (a) Can 3 divide o(a)? Either…

A: Given that G is a finite group such that ∀a∈G, o(a)≠3 (a) We have to determine whether 3 divides…

Q: 2. a) The operation in a finite group can be specified by writing down a table. Write the table for…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: This is abstract Algebra:   Suppose that G is an Abelian group of order 35 and every element of G…

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Q: Which of the following statements are True, and which are False ? Give reasons for your answers. (a)…

A: Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you.…

Q: If G is abelian group and (m,n € G ) then n'mn= m

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Q: 18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic if and only if…

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Q: G is a group with identity element e.  If a and b are elements of G, and n is any integer, show…

A: Let if n=0, the statement is,

Q: 1. Let (G, *) be a finite group of order n. Then (a). Let n = pq (p, q are a prime numbers) and let…

A: This is a problem of Group Theory.

Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

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Q: For each of the following, write TRUE if the statement is always true. Otherwise, write FALSE. (a)…

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Q: Which one of the following is true? a) Every abelian group is cyclic. b) Every group of order 2 has…

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Q: Consider the following statements B. Every cyclic group is abelian C. Every abelian group is cyclic…

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

Q: 3) Let G be a group. Show that if Aut (G) is cyclic, then G is abelian.

A: Solution is given below

Q: 4) Let set G be the Cartisian product of G1 × G2, where G1 and G2 are 2 groups. Define a binary…

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Q: Which of the following is not a group? * The set of non-zero real numbers under division. The set of…

A: We will use definition of group

Q: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: *…

A: Let G be a finite group and 'a' be an element  in G  s.t O(a)=9

Q: Determine whether the binary operation * gives a group structure on the given set. If no group…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 1. There can only be one group of order h, when h is a prime number (T) or_(F) 2. There is only one…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: If G is a finite group with |G|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given G&lt;160 It have the subgroups of order 10 , 16 , 20 Hence G must be the common multiple of 10…

Q: Q3: Prove or disprove any four of the following 1.Any abelain group is simple.

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Q: 1. - your answers by either writing a sentence or two or by providing a counterexam- ple. Determine…

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Q: 12

A: We can show this as shown below:

Q: Which of the following is a group? The set {1,3,4} under multiplication modulo 5. The set of…

A: Second option is correct.

Q: Which of the following statement is false? The symmetric group S has 10! elements. O Every group is…

A: (1) Given that the group S10 has 10! elements. The statement is true because Sn have n! elements.…

Q: 1. This is an exercise from math100a which gives us a characterization of cyelic groups. (a) Suppose…

A: It is given that

Q: (c) Show that Proposition 2 no longer holds without the condition that G is commutative. That is,…

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Q: Which of the following is a group? O The set of even integers under addition. O The set {1,3,4)…

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Q: 5.30. Let G be a finite abelian group. Suppose that, for every n e N, there are at most n elements a…

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Q: Determine which of the following sets G, with the indicated operation is a group. Which is an…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Prove or disprove the following statements: (a) If (ab)n+1 = an+¹fn+¹ a, b = G, n=1,2,..., then G is…

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Q: 2.6 Consider the integer group G (0, 1. 2. ,31) under modulo-32 addition. Show that H = (0, 4, 8,…

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Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.