Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E) < G, i.e. the preimage of E is a subgroup of G. Prove that if E < H, prove that y-(E) < G.

Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

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 nonabelian group. If H and K are cyclic subgroups of G, does it follow that H ∩ k is also…

A: Given that G is non abelian group. Given that H and K are cyclic subgroups of G. The cyclic group…

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

A:

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

A:

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: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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: 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: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

A:

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

A:

Q: If N and M are normal subgroups of G, prove that NM is also a normalsubgroup of G.

A: Given N and M are normal N and M are normal subgroup of G. We have to prove: NM is a subgroup of G…

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: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

A:

Q: . Let H be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

A:

Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

A:

Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

Q: Let Ha normal subgroup of G. Show that if x.v EG Such that xyEHthen yxEH- be a group and Attach File…

A:

Q: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

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: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: Let G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

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 H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

A:

Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

A:

Q: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

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: 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 C(a) is a subgroup of G.

A:

Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

A:

Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

A:

Q: If op is a homomorphism of group G onto G with kernel K and Ñ is a normal subgroup of G. N = {x E G|…

A: What is Group Homomorphism: If there exists a bijective map θ:G→G' for two given groups G and G',…

Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

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: Let be a group and Ha normal subgroup of G. Show that if y.VEG such that xyEH then yx EH

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: Let H be a subgroup of G. If G has exactly one subgroup of order |H|, then show that for all g e G,…

A:

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

A:

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

A:

Q: |Let G and H be groups, and let 0 : G > H be a homomorphism. The set {x E G| 0(x) |e} is called the…

A: From the given information:Let G and H be the groups

Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

A:

Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

A: