Q- (Subgroup Test for Finite Groups). Let G be a finite group. Prove that a nonempty subset H ⊆ G is a subgroup of G if and only if H is closed under the group operation of G.

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

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Q: Let G be a cyclic group with more than two elements: 1. Prove that G has at least two different…

A: Given: Let G be a cyclic group.

Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

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: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

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Q: Prove that a group of order 12 must have an element of order 2.

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Q: Let G be a cyclic group and H be a subgroup of G. Prove that H is a normal subgroup in G and G/H is…

A: Given:- H be a normal subgroup of a group  G⇒gHg-1=H∀g∈G Also given GH  is a cyclic and we need to…

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

A: Solution:-

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of the order…

A: Here given that G is a finite Abelian group that has exactly one subgroup for each divisor of the…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all…

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Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

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Q: Let G be an abelian group. Show that the set containing all elements of G with finite order is a…

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Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |

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Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

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

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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: Suppose that G is a finite group and that Z10 is a homomorphicimage of G. What can we say about |G|?…

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Q: Show that a finite group of even order that has a cyclic Sylow 2-subgroup is not simple

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Q: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.

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Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for eah divisor of |G|.  Show…

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Q: Suppose that G is a group of order 105 with the property that G hasexactly one subgroup for each…

A: Given G is a group of order 105 .

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: 10. The set of automorphisms of a group is a group under the operation of function composition. True…

A: Isomorphism : A bijective mapping  that preserves binary compositions in both groups is called…

Q: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: A proper subgroup H in a group G is called a maximal subgroup if there is no proper subgroup K of G…

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Q: Let G be an Abelian group and let H be the subgroup consisting ofall elements of G that have finite…

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Q: Let Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

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: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False

A: we will give the counter example in support of our answer.

Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: (iv) Does there exist a group G such that [G, G] is non-abelian? Give an example, or prove that such…

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Q: Let G be a abelian group. Prove that the set of all elements of finite order forms a subgroup of G.

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

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: 3. Let G be a finite group with an odd number of elements. Prove that G is isomorphic to a subgroup…

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Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of |G|. Show…

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Q: i have included a picture of the question i need help understanding.thank you in advance. please…

A: Let H and K are two subgroups of the group G.To show

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

A: According to the given information it is required to prove that every group of order 375 has a…

Q: 10. Prove that all finite groups of order two are isomorphic.

A: Here we use basic definitions of Group Theory .