Suppose G is a cyclic group with an element with infinite order. How many, elements of G have finite order? Explain your answer. P8 57 Dopós 8.

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: Is the set of numbers described below a commutative group under the given operation? Even integers;…

A: A group is a set with binary operation that satisfies the following properties a) It is closed under…

Q: If G is a finite group with |G|<180 and G has subg then the order of G is: * O 60 O 90 О 30 O 80…

A: Order of G is 90.

Q: An is the alternating group on n letters. What is |An|?

A:

Q: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

A: Cyclic groups and order of an element

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

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: (d) Is a21 a generator of G? Why or why not? (e) Let H < G. If |H| = 7, then which cyclic subgroup…

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Q: 7.30. Show that there are no simple groups of order 56.

A:

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: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

A:

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

A: A detailed solution is given below.

Q: 9. Which of the following groups are cyclic? Justify your answer. a) Z;×Z12 b) Z1,×Z, c) Z2×Z8×Z16…

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Q: 5. Find the number of generators of the cyclic group Z15

A: To find the number of generators of the cyclic group ℤ15.

Q: 1. Show that the set {5, 15, 25, 35] is a group under modulo 40. What is the identity element of…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: Show that if n > 6, then the symmetric group Sn contains an element of order greater than n

A: According to the problem, we have

Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: 3. Suppose that a group G contains elements of orders 1, 2, 3, 4, ..., 8, 9. What is the minimum…

A:

Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

Q: many

A: We have to find the number of generator of the given group of order

Q: 1.Show that the set {5,10,25,35} is a group under multiplication modulo 40 by constructing its…

A: Let us denote the operation given in the question, multiplication modulo 40, with · and the usual…

Q: Give an example of a group with 105 elements. Give two examples of groups with 44 elements.

A: It is known that the set Z105 = {0, 1, 2, 3 ….., 104} forms a group under addition modulo and…

Q: Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of…

A: Definition of subgroup: Let (G ,*) be a group and H be a subset of G then H is said to be subgroup…

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

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Q: How many elements of the cyclic group GF(81)* are generators?

A:

Q: 4. What's mean by a group? Is every group commutative? Explain your answer.

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Q: What is a quotient group and conjugacy class

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Q: Show, by example, that in a factor group G/H it can happen thataH = bH but |a| ≠ |b|.

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

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Q: 4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of…

A: To list the possible orders of the subgroup generated by a and b under the given conditions

Q: Which among is not a subgroup of a cyclic group of order 12? (a*) (a³) O. O Option 4 (a²) (a*) O-

A: We have to check

Q: Every commutative group has at least element ??

A: Every commutative group has at least               element ?  We know that , every commutative group…

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

A:

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

A:

Q: 17. Determine whether the set of all odd integers with mul- tiplication as the operation is a group.…

A: 17. The objective is to determine whether the set of all odd integers with multiplication as the…

Q: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…

A:

Q: How much cyclic groups does have U(15

A: Cyclic group  is a group that is generated by a single element.

Q: (c) Suppose that G = (a), a e, and a5 = e. Construct a Cayley table for the group (G,.). CIG [11

A: We shall answer first question only as you have asked more than one different question. For others…

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10…

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

Q: Find the order of the group and the order of each element in the group. In each case, how are the…

A: The no. of elements present in a group is the order of the group. n is the least positive integer…

Q: True or False:  No group of order 21 is simple.

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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: Prove Corollary 36.4 which says, "Let G be a finite group. Then G is a p-group if and only if |G| is…

A: First suppose that G is a p-group. Let q is a prime which divides |G|, then G contains an element of…

Q: Find a set of generators and relations for Q8

A: Given: The Quaternion Group Q8. To find: The set of generators and relations.

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: Show that there is no simple group G of order 33, 34, 35, 38, 39, 42, 44, 46, 50, 51. Show that…

A:

Q: Is the set of numbers described below a group under the given operation? Integers; addition Yes O No

A: Given: A set of integers under the addition operation. We have determine whether the set of integers…

Q: 2) Given example of an infinite group in which every nontrivial subgroup is infinite.

A: Let G=a be an infinite cyclic group generated by a, whose identity element is e. Let g∈G, g≠e,…

Q: (a) What element in G is a2022? (b) What is the inverse of a$3? (c) Which elements in G have order…

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Q: (6) Suppose that G = ( a) and |G| = 140. (a) What element in G is a2022? (b) What is the inverse of…

A:

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.