If H is normal, then xHxH KCH. Since o(K)= o(H), then we obtain H = K, this shows that H is unique. Sylow's Third Theorem. The number of Sylow p-subgroups in G, for a given prime p, is of the form 1+kp, where k is some non-negative integer and (1 + kp) | o(G). Proof. Let M be the set of all Sylow p-subgroup of G and H be any fixed number of

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

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Q: Problem 1. Suppose a and b are integers. Show that the subgroup of (Z/nZ, +) generated by [a]n…

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Q: The index of [G:H] is a prime number that divides H and divides G O the double number of left or…

A: The double number of left or right cosets of H in G.

Q: Q3: Recall that Theorem 3.9.11 (Lagrange's Theorem) states that if H is a subgroup of G, then o(G)…

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A: This is a good exercise in working with cosets. We first find out the subgroup $H$ and then working…

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Q: Question 2. The subgroup generated by S could have been defined a second way, as the set of all…

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Q: (a) In S4, find the subgroup H generated by (123) and (23) (b) For o = (234), find the subgroup oHo

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Q: Let a= (123)(145) in S6. Write a" in disjoint cycle form. None of these (32541) O (14523) O (16)(24)

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

Q: (4) Let N and H be subgroups of G and let N 4G. Prove that NH is normal in H.

A: Note: Since you have posted a multiple question according to guidelines I will solve first…

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Q: Which of the following are group homomoprhisms? (i): : Z₂ → Z₂, x + x² (mod 2). (ii): o: Z→ Z₂, x²…

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Q: et G be S, x U8 and H proper normal Subgroup of G. Then write Coyleys Toble Zor the fector roup G/H.…

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Q: S={x element of Z is even} as a subgroup of Z. What crucial elements are in S make this happen?

A: Given S={x element of Z is even} ={..., -4z, -2z, 0,2z, 4z,...} Since O£Z because 0 consider as a…

Q: Which of the following is correct? * O None of these O U(14) is isomorphic to U(7) O U(14) is…

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A: Consider the given information.

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Q: Q. Let G be a group, and let p be a prime. Prove that |G| = p" for each a E G, for some integer n >0…

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Q: Find the index of H = {0,3} in Z. 1 2 Find the left cosets of the subgroup 4Z in 2Z. A. {2Z} B. {4Z}…

A: H={0,3} is subgroup of order 2 of group Z6 which has order 6  So, index of H = {0,3} in Z6 is (order…

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: Let G = (Zp, +) and show that G has no nontrivial proper subgroups

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Q: Please solve and show working. Q: If a nonabelian group G has order 100, show that G has more than…

A: Sol:- We can use Sylow's theorems to approach this problem. Recall that if G is a finite group and p…

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A: We proved by contradiction that H is normal in G  The procedure is given below

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Q: 1, and let ø denote Let m and n be positive integers with gcd(m, n) Euler's o-function (Definition…

A: Given: Let's determine the following a) Let's verify that (a,b) is a generator for the group ℤnℤ×ℤmℤ…

Q: x = {[8 -d]|a,b€R} Prove that X*: =X\ {[%]} is a subgroup of GL(2, R).

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Q: Q3: Prove or disprove any four of the following 1.Any abelain group is simple.

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

Q: (8) Let n > 2 be an even integer. Show that Dn has at least n/2 subgroups isomorphic to the Klein…

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Q: Prove or Disprove that the Klein 4-group V4 is isomorphic to Z4.

A: The Klein 4 Group is a least non cyclic group. All the none identity element of the Group, which…

Q: Does Sn have a subgroup isomorphic to An, other than An (n ≥ 5).

A: A very famous result. Lot of techniques and theories about A_n will be used. The long solution has…