51. Let N be a normal subgroup of G and let H be a subgroup of G. If N is a subgroup of H, prove that H/N is a normal subgroup of G/N if and only if H is a normal subgroup of G.

Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: 3 Let N be a normal subgroup of G. Suppose Nn [G,G] = 1. Show that N < Z(G). group. -] Show that…

A: Before we start, let's define the terms:Normal subgroup: A subgroup N of a group G is called a…

Q: 16. If H is a subgroup of G, and K is a normal subgroup of G, prove that HK = KH. 1

A: Proof:Let H be a subgroup of G and K be a normal subgroup of G.We will proof HK=KHLet, hk be an…

Q: Exercise 7.25. A subgroup H of a group K is called a characteristic subgroup of K if ø(H) = H for…

A: A subgroup H of a group K is called a characteristic subgroup of K if ∅(H)=H for all ∅∈Aut(K) Let H…

Q: Prove that if N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal…

A: To Prove If N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal subgroup…

Q: 11.8 Let N be a normal subgroup of G and let H be any subgroup of G. Let NH= {nh | ne N, h € H).…

A:

Q: Let N be a normal subgroup of G and let K/N be a normal subgroupof G/N. Prove that K is a normal…

A:

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: 1. Let G and G' be two groups. Let p: G G' and :G G' be two homomorphisms then prove that H = (x EGI…

A: Using the definition of subgroups, we will first show that H is a subgroup of G. And then we will…

Q: a. Prove or Disprove. If H is an abelian normal subgroup of G then H be contained in Z(G).

A:

Q: 2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a…

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Q: a) Show that a group of order 700 has a normal subgroup of order 175.

A: To show that a group of order 700 has a normal subgroup of order 175, we will make use of the Sylow…

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…

Q: (Third Isomorphism Theorem) If M and N are normal subgroups ofG and N &lt;= M, prove that…

A: Given: M and N are normal subgroups of G with N≤M.We want to prove that G/N/M/N is isomorphic to…

Q: 4. Let G be a group of order 14. Using Lagrange's theorem, show that every subgroup of G either…

A:

Q: Let in S3 let H = {(1), (12)}. Show that (13)H(23)H ≠ (13)(23)H.(This proves that when H is not a…

A: Given: H = {(1), (12)} We need to prove that (13)H(23)H ≠ (13)(23)H.

Q: If G is a group and H is a subgroup of index 2 in G show that H is a normal subgroup of G.

A:

Q: Let N₁ and N₂ be distinct index 2 subgroups of G. Prove that nN₂ is a normal subgroup of G and that…

A: Given problem is from group theory. To prove this result I have used fundamental theorem of…

Q: 1. Let G₁ and G₂ be two groups, with identities e₁ and e2, respectively. Let G = G₁ x G2 and let H…

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Q: Q3) Prove or disprove 4. Any non-trivial group has at least 2 normal subgroups.

A: Here we have to prove that any non trivial group has atleast 2 normal subgroups.

Q: Let G be a group of order 108. Show that there  exists a normal subgroup of order 27 or 9.

A: Given that G be a group of order 108 then we have to prove that there exists a normal subgroup of…

Q: Let G=H×K.If N is a normal subgroup of H and L is a normal subgroup of K,show that N×L is a normal…

A: As we know, e∈N and e∈L, then (e,e)∈N×L. If (n1,l1),(n2,l2)∈N×L, then…

Q: Modern Algebra: Give a counter-example to the statement (1) for a non-normal subgroup N. (1) Let P…

A: Given, P be a Sylow p-subgroup of a finite group G and N is a normal subgroup of G.  To prove:  P∩N…

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

A: Let G=S3 group of permutations on three elements. H=1 2 subgroup generated by 1 2.

Q: Let P has order 8 is a normal subgroup of G. Show that the order of automorphism of P (Aut(P)) is…

A: We will need some facts regarding the group A_4

Q: 3. Let A and B be groups. Show that S := {(a, e) | a € A} is a normal subgroup of G:= A x B and the…

A:

Q: If N is a normal subgroup of G and H is any subgroup of G, show  that NH is a subgroup of G.

A: If N is a normal subgroup of G and H is any subgroup of G, we have to show that NH is a subgroup of…

Q: If G is a group and H is a subgroup of index 2 in G show that H is a normal subgroup of G.

A:

Q: 12. Let : G→ H be a group homomorphism. suppose that K is a normal subgroup of G. Prove or disprove:…

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Q: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

A:

Q: Prove that if H is a subgroup of a finite group G of index p, where p is the smallest prime that…

A: This question is about group theory.

Q: Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal…

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

Q: The intersection of any two normal subgroup  of a group G is a normal subgroup of G.

A: We have to prove that the intersection of any two normal subgroup of a group G is a normal subgroup…