Prove that a normal subgroup must be a union of conjugacy classes.

Q: Let (H,*) be a subgroup of a group (G,*) the relation of congruence modulo a subgroup H, .=. mod H…

A: (H,*) be a subgroup of G. For a,b ∈G, we say that a≡b(mod H) (a is congruent to b modulo H) ifab-1∈H…

Q: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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

A: We will prove the statement.

Q: Show that the intersection of two normal subgroups of G is a normalsubgroup of G. Generalize

A: Given: To prove the intersection of two normal subgroups of G is a normal subgroup. Consider G be a…

Q: Prove or give counter example Every characteristic subgroup is fully invariant

A:

Q: Show that no proper subgroup of S4 contains both (1,2,3, 4) and (1,2).

A: We have to show that no proper subgroup of S4 contains both 1,2,3,4 and 1,2. We know that the…

Q: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

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

A: Solution:-

Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Suppose that H is a proper subgroup of Z under addition and that Hcontains 12, 30, and 54. What are…

A: It is given that H is a proper subgroup of Z under addition and that H contains 12, 30 and 54.

Q: The intersection of two normal subgroups is always normal sub group. (True/False)

A:

Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

Q: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

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Q: Prove that a kernel of a homomorphism is a subgroup. 52.

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Q: Show that Z has infinitely many subgroups isomorphic to Z.

A: We have to show that Z has an infinitely many subgroups isomorphic to Z.

Q: Show that S = SU(2) contains a subgroup isomorphic to S'.

A: Let's define S1 as the set { (x,y)∣ x2+y2 = 1 } ⊂ R2 we may think of S3 as S3={ (a,b) ∈ C2:…

Q: Show that a subgroup of a solvable group is solvable.

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Q: Construct a subgroup lattice for the group Z/48Z.

A:

Q: Let n be an integer greater than two. Show that no subgroup of order two is normal in Sn.

A: To prove that no subgroup of order 2 in the symmetric group Sn (n >2) is normal.

Q: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

A:

Q: The set of all odd permutation subgroups in S, form

A: False,

Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

A:

Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

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

A:

Q: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

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: 37. Is the set Ze of even integers a subgroup of Z with addition? What's about (Z,,+)?

A:

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

A: Solution

Q: H. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.

A:

Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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Q: Prove that any group of order 75 can have at most one subgroup of order 25.

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Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: prove that the set of even permutations in sn forms a subgroup of sn

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Q: Show that the only Normal subgroup of S3(All permutations of 3 distinct elements) is the subgroup…

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Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

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Q: If N is a normal subgroup of G and G/N is abelian. Then G is also abelian. Select one: True False

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Q: a subgroup of (G, *) , known as the normalizer subgroup of (H, *) in (G, *).

A:

Q: Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.

A: A subset H of G is called a subgroup of G if H also form a group under the same operation.

Q: (a) Prove that any subgroup of Sn contain either even permutations only or equal number of even and…

A: As per guidelines we are allowed to solved one question at a time so i am solving first one please…

Q: think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal…

A: A subgroup H of h is called normal subgroup of h if θH⊆H  ∀θ∈AutG

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

A:

Q: Determine if B is a subgroup of A

A: Introduction: Subgroup is a part of a group. More precisely, a subgroup is a non-empty subset of a…

Q: Find the subgroup of ℤ generated by the subset {6,9}.

A:

Q: Show that every group of order 56 has a proper nontrivial normalsubgroup.

A: