Show that the center of a group of order 60 cannot have order 4.

Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: Prove that a group of order 375 has a subgroup of order 15.

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Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: Prove that every group of order 330 is not simple.

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

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: Prove O3 is not a group

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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: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

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

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Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

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Q: Let G be a group of order 60. Show that G has exactly four elementsof order 5 or exactly 24 elements…

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Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

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

A: using  third Sylow Theorem

Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: Z, x is not a group.

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Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Consider find Subgraup. Dihedral group D- of order 2,3,4 and 6.

A: A group G of two generators x and y of order n and 2 respectively with some relation is called the…

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: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: = Prove that, there is no simple group of order 200.

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Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

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: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

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

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Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.

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: What can you say about the centre of a simple group?

A: 9

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

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

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

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

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Q: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution