1. Prove that if G is a group with prime order, then G is cyclic

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

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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…

Q: Suppose that a is a group element and a^6 = e. What are the possibilities for |a|? Provide reasons…

A: It is known that if G be a group and a be any element of group G, then n  is said to be order of a…

Q: a. Prove that the set of numbers {1,2, 4,5, 7,8} forms an Abelian group under multiplication modulo…

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Q: a(H n K) Let G is a group, H, K C G, and a e G. Is it the case that aH N aK? Provide a proof or…

A: According to the given information,

Q: 1. Consider the group D5. Give five examples of D5 acting on a set, other than the trivial action.…

A: Given: Consider the group D5. Give five examples of D5 acting on a set, other than the trivial…

Q: True or False. Every group of order 159 is cyclic.

A: According to the application of the Sylow theorems, it can be stated that: The group, G is not…

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: mat do you call a group that is commutative?

A: A detailed solution is given below.

Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m> 1, the…

A: Since part b,part are independent of part a,as per the guidelines I am answering the part a only.…

Q: Prove that a group of even order must have an element of order 2

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Q: Abstract Algebra Check whether or not a group of order 407 is simple.

A: A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1,…

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Q: If G is a group then Z(G)◄G. *True False

A: Use the relevant definitions.

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

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Q: 22, Use mathematical induction to prove that if a1, a2, ... , an are elements of a group G, then…

A: See the detailed solution below.

Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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Q: 2) (b*a)1 = . If a, b are elements of a group G?

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Q: Classify groups of order 2p as best as you can. Give a proof for your assertions and, when…

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Q: Consider the group (U(20), '20). 1. What is the order of the group? 2. What is the inverse of 7,9…

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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: 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: 2. Let G be a group. Pro-

A: Let G be a group .

Q: Abstract Algebra:   Prove or disprove, as appropriate: If G and H are cyclic groups, then G x H is a…

A: If the direct  product of two groups is cyclic, does it follow that each of them is cyclic?

Q: Which of the following is a group? O O

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Q: (a) Prove that: If G is a group, a,b € G, |a|=5 and a^3b-ba^3, then ab-ba

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Q: Is the set of positive integers a group under the operation of addition? Is the set of positive…

A: A set is said to be group on a binary operation if  i) It is closed ii) It is associative iii) There…

Q: Give an example or explain why the following is not possible.  An infinite group that is finitely…

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Q: Let a,b be elements of a group G .Assume that a has order 5 and  a^3b=ba^3. prove that ab=ba.

A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.

Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

A: Order of element a ∈ G is the smallest positive integer n, such that an= e, where e denotes the…

Q: Give an example, with justification, of a semi-group which is not a group.

A: A semigroup defined on a set is a binary operator which satisfies the association property of…

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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Q: Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic

A: The proof has if part and only if part of the proof. If part: We are given that  a finite group is…

Q: Which of the following is the only trivial sub- group of a group G = {e, a,b, c}? {e,b} {e, a, b, c}…

A: We have to check

Q: True or False? Prove or disprove as appropriate: Suppose G is a group, H ≤ G, and a, b ∈ G. If aH =…

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Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a+bv2 : a,…

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Q: Which of the following is the only trivial sub-group of a group G = {e, a, b, c}? {e}, {e, a, b, c},…

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Q: Suppose G is a cyclic group with an element with infinite order. How many, elements of G have finite…

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Q: Is the set of numbers described below a group under the given operation? Rational numbers;…

A: This is a question of abstract algebra.

Q: TASK 2: Prove that if G is a group and a, b = G, then (1) o(a¹ba) = o(b) o(ab) = o(ba)

A: These two results follow by proving G is abelian. We present the detailed solution in the next step.…

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: State whether the following statement is true or false: A group may have two or more identity…

A: A group have unique identity.

Q: 18. Let peR.o G = Show that G is a group under matrix multiplication.

A: Given: G=a00a|a∈ℝ,a≠0 We need to show that G is a group under matrix multiplication.

Q: et G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Since H is a cyclic group. and K is cyclic group. H=a,K=b If H⊂K then H∩K=H where H is cyclic If K⊂H…

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: プーー Let G be a group. H,ksG Sot H= for same a,beG. That is k are cyclic subgroup.of H cund G.Does.…

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Q: 7. Let G be a group. Then Z(G) is always of smaller size than G. *True *False

A: We know that ZG=a∈G  ag=ga ∀ g∈G which means that ZG is the set of those elements of the group G…