Check whether a group of order 156 is simple or not.

Q: Compute the factor group (Z6 x Z4)/(([2]6, [2]4))

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Q: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

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Q: · In a group, prove that (ab) = b-'a-|

A: As you asking for question number 7 , I solve for you.

Q: Define Group theory ?

A: To define group theory

Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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Q: How many elements of a cyclic group with order 10 have order dividing 5?

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Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

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

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

Q: Construct the group table for A4.

A: S4 is the set of all permutations on the set of four elements namely {1,2,3,4}. Then A4 is the set…

Q: 10. Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not…

A: Given that,   (a) Group of order 40. Let G=40,  i.e, G=40=23.5, Since, 5-Sylow subgroup, i.e…

Q: Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176, 189, 195, 200 is not simple.

A: To show that the group of order 45 is not simple.

Q: Show that the set {5, 15, 25, 35} is a group under multiplication module 40.What is the identity…

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

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

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Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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Q: How many ways can a group of five be chosen to work on a project? As in Example 9.5 4, since the of…

A: Given that, the computer programming team has 9 members.

Q: Prove that the set of natural numbers N form a group under the operation of multiplication.

A: The set N of all natural numbers 1, 2, 3, 4, 5... does not form a group with respect to…

Q: 1. Show that every group of prime order is simple.

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Q: How many elements of a cyclic group with order 14 have order 7?

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Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

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Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: Show that every group of order < 60 is solvable.

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Q: Let α,β ESs ( a = (1,8,5,7)(2,4) and B= (1,3,2,5,8,4,7,6). Compute aß. Symmetric group) where

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Q: Prove that any group of order 189, 195, 200 is not simple. 84/135, 140. 84, 135, 140, 165, 175, 176,

A: Sylow's test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1…

Q: Show that there are no simple groups of order 255 = (3)(5)(17).

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Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Prove that there is no simple group of order 528 = 24 . 3 . 11.

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Q: The order of identity element equal itself in any group

A: .

Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

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Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

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

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Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: (i). There is a simple group of order 2021.

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Q: Show that the groups Z8xZ20xZ12 and Z120xZ4xZ4 are isomorphic by define a one-one and onto map? what…

A: We will use the basic knowledge of groups and abstract algebra to answer this question.

Q: The group (Z6,6) contains only 4 subgroups

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Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

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Q: List all abelian groups (up to isomorphism) of order 600

A: To List all abelian groups (up to isomorphism) of order 600

Q: Any group of order 520 is simple

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

Q: Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.

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Q: The group (Z6,+6) contains only 4 subgroups

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Q: What is the numbers group of

A: The given number is 5. The value of 5 is 2.23606...

Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…

A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…

Q: Exercise 3: Prove that every element of a finite group is of a finite order.

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Q: 6.7 Construct a nonabelian group of order 16, and one of order 24.

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Q: (5) Make a list of all group homomorphisms from Z4 → Z8.

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Q: G1 = Z÷ and G2 = Z, *

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Q: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.