Let G be a group of order 60. If the Sylow 3-subgroup is normal,show that the Sylow 5-subgroup is normal.

Q: consider S3 and N = { (1), (12)} and H = { (1) , (123), (132)} both are subgroups of S3. Determine…

A:

Q: Let G (a.) Let a = (123) (14). Write a ¹2 in disjoint cycle form. (b.) Consider the subgroup H =…

A:

Q: 4. Let H & K are two subgroups or a group G such that H is normal in G then show that HK is a…

A:

Q: Solution of the differential equation 9 + = y² x dy x is %3D dz

A:

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

A:

Q: What can you say about a subgroup of D3 that contains R240 and areflection F? What can you say about…

A: Given:  Subgroup of D3 that contains R240 and reflection, F.             Subgroup of D3 that…

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

A:

Q: Let G = S4. Find a subnormal subgroup H of G such tht the normalizer of H is a Sylow 2- subgroup of…

A: Let's breaking down the problem step by step:1. G=S4 is the symmetric group on 4 elements, which has…

Q: The normalizer N(a) of a∈G is a subgroup of G.

A:

Q: Let G be a group and H a normal subgroup of G. Show that if x.V EG Such that xvEH then X,y xyƐH yx…

A: The solution is given as

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

A: We have to find all the Sylow 3-subgroups of S4.

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

A:

Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

A:

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4.

A: It is given that a Sylow 2-subgroup of S4. So, we have, S4=4!=24=3×23. let k be a Sylow 2-subgroup…

Q: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

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 G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

A:

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

A:

Q: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

A:

Q: Let K={β∈S5∣β(1)=2 and β(4)=4}K={β∈S5∣β(1)=2 and β(4)=4}. Prove that KK is not a subgroup of S5.

A:

Q: subgroup of D12. Prove that D6 is

A:

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: Let G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

A:

Q: In S3, find an example of an H < G, and g1, g2 ∈ G such that the two conjugate subgroups are not…

A: Let H be a subgroup of G Then a H1={ghg | g,h belongs to G,H respectively}. is a conjugate subgroup

Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin square…

A: G be a group of order 3 Consider G=e,a,bwhere e is the identity then the Cayley table is…

Q: Let G be a group of order 2p, where p is  prime. Show that G has a normal subgroup  of order p.

A: G be a group of order 2p, where p is prime.    Claim : G has a normal subgroup of order p.

Q: Find all subgroups of Z60 and draw a lattice diagram for them.

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: Let H and K be normal subgroups in G such that H n K = {1}. Show that hk = kh for all he H and k e…

A:

Q: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: Find the partition of Ze into cosets of the subgroup H = {0,3}.

A: In this problem we have to find the partition of  into cosets of the subgroup .

Q: Let A be a subset of the group G. Prove that the normalizer of A, NG(A) = { g e G : gAg-1 = A}, is a…

A:

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: Answer the followings: 1. Let H = {[a b]: a, b, d € R, ad # 0}. Is H a normal subgroup of GL₂(R)? 2.…

A:

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

A: