5. Show that the intersection of two normal subgroups of G is a normal subgroup of G 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: 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 of order p, q, where p, q are primes and p < q. Prove that (a). G has only one…

A: It is given that G, * is a group of order p·q where p, q are primes and p&lt;q. Show that G has only…

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

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: 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: If K is a normal subgroup of a finite group G and S is a Sylow p-suby

A: Given that if K is a normal subgroup of a finite group  G and S is a Sylow p-subgroup of G. then K∩S…

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: 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: (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 &gt;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: Theorem: Let (K,+)is a subgroup of a group (H, ) and (H,) is a subgroup of a group (G,) then (K, )is…

A:

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

A:

Q: 4. Let G, Q be groups, ɛ: G → Q a homomorphism. Prove or disprove the following. (a) For every…

A:

Q: 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…

A: According to our guidelines we can answer only first question and rest can be reposted. Not more…

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: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

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Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

A:

Q: Let G be a group and H, KG normal subgroups of G. Prove HnK≤ G.

A:

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: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

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: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

A:

Q: 4) Let H and K be a subgroup of a group Gif HAK,KAG and HAG then HnK is not normal subgroup of G.…

A: Given, H and K are subgroups of a group G. Also, given that H is normal in G and K is normal in G.…

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

A:

Q: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

A:

Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

A:

Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

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

A:

Q: How thata Show that an intersection of normal subgroups of a group G is again a normal subgroup of…

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Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

A:

Q: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.

A:

Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

Q: Prove that if H is a normal subgroup of G of prime index p. (Note G can be finite or infinite…

A: It is given that, H is a normal subgroup of G of prime index p. (Here G can be a finite or infinite…

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: Show that every subgroup H of the group G of index two is normal.

A:

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: H. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.

A:

Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

A:

Q: Prove that any subgroup H (of a group G) that has index 2 (i.e. only 2 cosets) must be normal in G

A: To show that H is a normal subgroup we have to show that every left coset is also a right coset. We…

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

A:

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

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: