7. Suppose G is a finite group and let n be a factor of |G|. Suppose N≤ G is the only subgroup of G of size n. Prove that N is a normal subgroup of G.

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

A:

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

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: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

Q: 5. Let H be a subgroup of a group G. Prove that aH Ha-1 is a bijective mapping from the set of all…

A: Let H be a subgroup of a group G. Let f be a function from the set of all left cosets of H in G to…

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: Sylow-5 subgroups-

A:

Q: 5. Show that the set of all even permutations forms a subgroup of the symmetric group (S This group…

A:

Q: (a) class(x)| = |G: N(x)| for all x € G. (b) If H is a subgroup of G, then the number of conjugates…

A: Given Information: is a finite group.Normalizer: , where .To Prove:a) .b) Number of conjugates of…

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: Suppose that G is a group of order pnm, where p is prime and p doesnot divide m. Show that the…

A: As per the question we are given a group G of order pnm, where p is prime and p does not divide m.…

Q: 4. Let G be a group, and let H be a subgroup of G. Show that if H contains the commutator subgroup…

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…

A:

Q: Show that if G is a finite group of even order, then there is an a EG such that a is not the…

A:

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: Exercise 64. Let G be a group and let H be a subgroup of index n. If G| ≤n!, then G is not simple.…

A: According to new guild line , we will provide solution of first question ,since you posted multiple…

Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: 4. If H is a subgroup of G, then show that the set W = ngHg¹ is a normal 9€G subgroup of G.

A: Given That : H is a subgroup of G  To Show: The set W=∩g∈GgHg-1 is a normal Subgroup.

Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…

A: See below

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

A:

Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

A:

Q: how that the set of Intergers is a Normal subgroup of the set of Real number

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 the integration by parts was used to integrate the product of e^ax and sin(bx) and cos(bx).…

A: 1. To evaluate the integral ∫eaxcosbxdx+i∫eaxsinbxdx:

Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in 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: 8. Show that if |G| = 12 then G has a normal Sylow 2-subgroup or a normal Sylow 3-subgroup.

A: Ans-In the given question we have to show  if |g| = 12 then g has a normal sylow 2-subgroup or 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: 3. Let G be a group of order pq where p, q are two distinct prime numbers. (a) Assuming that p < q…

A: Given a group G of order pq where p and q are two distinct prime numbers.   Used definition:  number…

Q: Suppose that G is a group of order pnm, where p is prime and p doesnot divide m. Show that the…

A: Given: G is group of order pnm where p is prime and P does not divide m. To show: Number of sylow…