H be a subgroup of G.

Q: If p is a prime, prove that any group G of order 2p has a normal subgroup of order p and a normal…

A: To prove that any group of order 2p has a normal subgroup of order p and a normal subgroup in g

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: (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: Prove that the centralizer of a in Gis a subgroup of G where CG (a) = { y € G: ay=ya}.

A:

Q: If H is a normal subgroup of G and |H| = 2, prove that H is containedin the center of G.

A:

Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

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: If N is a normal subgroup of G and G/N=m , show that xmN forall x in 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 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 N is a normal subgroup of G and |G/N| = m, show that x" EN for all x in G.

A: Given: N is a normal subgroup of G.

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

A:

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: . 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, prove that the center Z(G) of a group G is a normal subgroup of G.

A: Let G be a group. Consider the subgroup ZG=x∈G | ax=xa.

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: . 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 M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

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: Every subgroup of a group G is normal * False True

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: 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: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

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

A:

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

A:

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

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:

Q: Let H and K be two subgroups of a group G. Let HK={ab|a∈H,b∈K}. Then HK is a subgroup of G. true or…

A: F hv

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

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: 7. Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A:

Q: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

A:

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

A:

Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

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

A:

Q: Determine if B is a subgroup of A

A: Introduction: Subgroup is a part of a group. More precisely, a subgroup is a non-empty subset of a…

Q: Lemma 5 Let G be a group and Ha subgroup of G. Prove that the normalizer, Nc(H), is a 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: Let H be a subgroup of G and let a, be G. If Ha Hb, then* %3D aH = bH O a-1H = b-1H O Ha = Hb Ha-1 =…

A:

Q: Let H be a subgroup of G, define C(H) the centralizer of H.

A: