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: 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: 2.26. If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of…

A:

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: 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: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

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

A:

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

A:

Q: 10. Let G be abelian and let H be a subgroup of G. Show that G/H is abelian.

A: We have to prove that given theorem.

Q: Suppose there is a homomorphism Φ from G onto Z2 ⨁ Z2. Provethat G is the union of three proper…

A: Proof: 1. Since Φ is a homomorphism from G onto Z2 ⨁ Z2, we have that Φ(G) = Z2 ⨁ Z2. Let e denote…

Q: 1. Let G be a cyclic group of order n and k❘n. Show that G has a subgroup of order k; "

A: To prove that a cyclic group ( G ) of order ( n ) has a subgroup of order ( k ) for every ( k ) that…

Q: 53. Let H be a subgroup of G and C(H) = {g €G|gh = hg for all he H}. Prove C(H) is a subgroup of G.…

A:

Q: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let N be a normal subgroup of a finite group G.  Prove that the order of the group element gN in G/N…

A:

Q: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

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: Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n,…

A: Given, G is a finite group and H is a subgroup of G of order n.

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

A:

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 G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

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: B. Let G be a group of order 60. Is there exist a subgroup of G of order 24? Explain your answer

A: We have to give reason that is it possible a subgroup of order 24 of a group of order 60.

Q: 4. If p: G → G' is a homomorphism, prove that ker ý is a normal subgroup of G.

A: To prove that kernal is a normal subgroup of G

Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

A:

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: 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: Let G be a group with the operation * and let H be a subgroup of G. Show that if H and G/H have the…

A: To show that H is a normal subgroup of G, we need to demonstrate that for every element g in G, the…

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: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

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:…

A:

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: