Suppose G| = 170, PE Syls(G), and QE Sylı7(G). %3D (i) Calculate ns(G) and n17(G). (ii) Is P4G and Q4 G? Why? ii) Is PQ a subgroup of G? Why? iv) Calculate [G: PQ]; (write all details).

Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

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

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Q: Which of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?

A: Since 0 does not divides 12.

Q: Let n > 2 be an integer, and let X C Sn × S, be the set X = {(ơ, T) | 0(1) = T(1)}. Show that X is…

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Q: Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that H is…

A: We know that S3=1, 12, 13, 23, 123, 132. Giventhat H=12 is a subgroup of S3. H=1, 12We have to show…

Q: When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does…

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Q: If N is a normal subgroup of G and G/N=m , show that xmN forall x in G.

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Q: is a subgroup of Z1, of order: 3 12 O 1 The following is a Cayley table for a group G. 2. 3.4 = 2 3…

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Q: Find a subgroup of Z12 ⨁ Z18 that is isomorphic to Z9 ⨁ Z4.

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Q: Is the set {3m + v3ni|m, n E Z, b|m – n} the normal subgroup of the (C, +)group?

A: given :

Q: Determine which of the following is a normal subgroup SL(2, R) Z, None of them S3 GL(2, R)

A: Zn is not a sub-group but the subgroups of Zn are normal subgroups.

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: 3. am e H for every a E G. Let H be a normal subgroup of a group G, and let m = (G : H). Show that

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

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Q: Which of the following cannot be an order of a subgroup of Z12? 4 3 Option 4 12

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Q: ) What’s special about a normal subgroup?  b.) When we say xH = Hx where H is a normal subgroup of…

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Q: Q/ How many non-trivial subgroups in s3 ? a) 2 b) 3 c) 4

A: We know that S3 = (1) , (1,2) , (1,3) , (2,3) , (1,2,3) , (1,2,3) Thus the subgroups of S3 are given…

Q: Let n > 2 be an integer. Prove that An is a normal subgroup of Sn.

A: In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members…

Q: ] Let N be the subgroup of (Z15, +) generated by 3, that is N = . i- What is the order of N+ 2 in…

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Q: Find the three Sylow 2-subgroups of S4

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Q: Q7. Suppose that the index of the subgroup H in G is two. If a and b are not in H, then ab ∈ H.…

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Q: Q/ How many non-trivial subgroups in s3 a) 2 b) 3 c) 4 d) 5

A: The set S3 is given by  S3=e, 12, 13, 123, 132 The subgroups of S3 are e, e, 12, e, 13, e, 23, e,…

Q: Find four different subgroups of S4 that are isomorphic to S3.

A: If there exists an isomorphism between the subgroups they are called isomorphic ( Isomorphism is…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: Let H = be a subgroup of S3, then H is normal subgroup of S3 a) True b) False

A:

Q: Determine all normal subgroups of Dn of order 2.

A: Dn is generated by two elementsa & bwithan=b2=e , andba=a-1 bthenbak=a-k bBy induction and…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

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Q: Find a subgroup of order 4 in U(1000).

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Q: How many subgroups does Z/60Z have?

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Q: The group U(14) has: اختر احدى الجابات only 2 subgroups 4 sub groups 7 subgroups 6 sub groups

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Q: 8. Is Zg isomorphic to D4? What about Z4 and D4? Can you find a subgroup of D4 isomorphic to Z4?

A: Now we have to answer the above question .

Q: Q/ How many non-trivial subgroups in s, ?

A: For the given statement

Q: Find the center and the commutator subgroup of S3 x Z12-

A: Solution

Q: Show that in D8, is a normal subgroup of and is a normal subgroup of D8, but is not a normal…

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Q: Can Z121 have a subgroup of order 20? Explain.

A: Subgroup of order 20

Q: 2. Every group of index 2 is normal.

A: Given :   Every group of index 2 is normal

Q: If H and K are subgroups of G of order 75 and 242 respectively, what can you say about H N K?

A: Solution

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

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -6 + 2Z contains the…

A: 10 is the element in the right coset.

Q: Find all the Sylow 3-subgroups of S4.

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Q: Determine which of the following is a normal subgroup O GL(2. R) SL(2. R) O None of them Os. S,

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Q: Let T; = {o € S, : 0(1) = 1}, with (n > 1). Prove that T, is a subgroup of S,, and hence, deduce…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (1) Find all subgroupsof (Zs.+s).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Suppose that G is cyclic and G = (a) where Ja| = 20. How many subgroups does G have?

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Q: D. Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that…

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Q: Show that S4(a) has no normal subgroup of order three. (b) has a normal subgroup of order four.

A: To prove that (1) No normal subgroup of order 3 exists in S4 and (2) there does exist a normal…

Q: (b) For every normal subgroup N <Q, ɛ-'(N) is a normal subgroup of G. (Recall that e-(N) = {g E…

A:

Q: b. Find the center and the commutator subgroup of S2 x Z7.

A: Now we knew that  Z2 is isomorphic to S2. So it is commutative group. The center subgroup of G := S2…

Q: Suppose that | G|= 20. What're possible orders for a subgroup of G. 35.

A: We can find the subgroup of G as below.

Q: a) How many subgroups does (Z10,O) have? What are they? b) How many subgroups does (Z74,Ð) have?…

A: 2. a) Consider the group ℤ10, ⊕. The elements of the above group are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…