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:

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: 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: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

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: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…

A:

Q: Let G be a group and H, K are subgroups of G with HK=KH. Prove that HK is a subgroup of G.

A: Given that, G be a group and H, K are sub groups of G with HK=KH. Let x∈HK. Then x=hk for some…

Q: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

Q: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

A:

Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

A:

Q: Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

A: " Let G be a group and H ≤ G.The subgroup H is normal in its normalizer NG(H), this imply that NG(H)…

Q: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

A:

Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

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: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

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: If a cyclic group T of G is normal in G; then show t subgroup of T is a normal subgroup in G

A: Given: A cyclic group T of G is normal in G.

Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

A:

Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

A:

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

A:

Q: Let G be a group and let H be a subgroup of G with |G : H| = 2. Prove that H a G, that is, H is a…

A:

Q: 1- Prove that if (Q -(0),) is a group, and H = an, m e Z} 1+2m is a subset of Q - {0)}, then prove…

A: A subset H of a group G, · is said to be a subgroup of G, · if for any a,b∈H we have: a·b∈H a-1∈H…

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: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

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

A:

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

Q: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

A:

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Although (H,*) and (K,*) are subgroup of a group (G,*) then (H * K, ) may field to be subgroup of…

A:

Q: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

Q: 4. Let H be a subgroup of a group G. Show that exactly one left coset of H is a subgroup.

A:

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

A:

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: Let let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

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

A:

Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

A:

Q: Let let G₁ be A be of Suppose Subgroup index a group and a normal of finite G+₁ that H

A: We know that if G is a group and H is a subgroup of G and x is an element in G of finite order n. If…

Q: Although (H,*) and (K,*) are subgroup of a group (G,) then (H * K,*) may field to be subgroup of (G,…

A:

Q: a. If G is a group of order 175, show that GIH=Z, where H is a normal subgroup of G. b. Show that Z…

A:

Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

A:

Q: 2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of…

A:

Q: 1. Let G be a group and let H, H, .. H, be the subgroups of G. The ...

A: