Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

Q: Prove or give counterexample. Let G be a group and H ≤ G. The subgroup H is normal in its…

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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: Let H be a subgroup of R under addition. Let K = {2ª | a E H}. Prove that K is a subgroup of R*…

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Q: ts] Let G be a group, and let H be a subgroup of G. Show that if G is cyclic then H is also cyclic.…

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Q: 2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

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Q: Let G be a group and let H and K be normal subgroups such that HnK = {e}. Let : G→G/HxG/K be the map…

A: Let  be  a  group  and let and  be  normal subgroups   such  that .Let   be  the map We have   to…

Q: Let C(x) represent the centralizes of a. If x is an element of group G with the order of x be 5 (|x|…

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Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

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Q: Let A = {fm,b : R → R | m ± 0 and fm,b(x) : = mx + b, m, b e R} be the group of affine m b functions…

A: We are given two groups.  A={fm,b:ℝ→ℝ|m≠0 and fm,b(x)=mx+b,m,b∈ℝ} B=mb01 | m,b∈ℝ,m≠0. We need to…

Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…

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Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by f(x) =…

A: Given : Let G be a group and N is a normal subgroup of G and Let f be a mapping from G to H/N…

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 and N is a normal subgroup of G. Let f be a mapping from G to H/N defined br f(x) =…

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Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: 2. Let G be a group, and let A be a non-empty subset of G. Define Ca(A) = {g G | gag¹ = a for all a…

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Q: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

Q: Let G be a group and let H and K be normal subgroups such that HnK={e}. Let : G→G/HxG/K be the map…

A: The objective of this question is to prove that the kernel of the map φ: G → G/H × G/K, where φ(g) =…

Q: Let H and K be subgroups of a group G. (a) Define HK = {hk | he H, ke K}. Show that if K is normal…

A: We will solve all the three parts. Given that  H and  K are subgroup of G

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: Let G be a group and let H and K be normal subgroups such that Hnk= G/H x G/K be the map o(g) = (Hg,…

A: A mapping from a group (G,o) to a group (G',o') is called a homomorphismif f(aob)=f(a)o'f(b)where a,…

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

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Q: Question 10. Prove in detail that G(T) is a subgroup of G.

A: Here we use one step subgroup test

Q: (3) For any group G, Z(G) ≤ [G, G].

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Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined b f(x) =…

A: Given :  G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by…

Q: Let G be a group. Define Z(G) = {x G | xg = gr for all g € G}. Prove Z(G) ≤ G. Find Z(S3) and Z(D4).

A: Solution:Let  G be a group. Define Clearly, eSuppose be  arbitrary.Then,  [since, x,y]Thus, 2nd…

Q: (c) Show that the Euclidean group E(2) is generated by the reflec- tions. More precisely, show that…

A: The Euclidean group E2 is generated by the reflections. That is any isometry of ℝ2 is a product of…

Q: 1. Let G be a group. Define the center of G by C(G) = {x € G: xg = Prove that C(G) is always a…

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Q: If H≤G and let C(H) = {x element G| xh=hx for all h element H} prove that C(H) is a subgroup of G.

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Q: ¹) Let H be a subgroup of a group G. Define NĠ (H) : = {g €G | g Hg¯¹ = H}. Prove that N(H) is a…

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: Find the distance of the point (1,3) from the line 2x-3y+9=0 measured       along a line x-y+1=0.

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Q: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

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Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

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Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: 8. Consider lincar multistep methods applied to stiff ODES. (a) Why are explicit methods typically…

A: Part(a):  We consider the linear multistep methods applied to Stafford's. First we explain why…