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. a re nto] Assume that G and H are both groups and that f:G →H is an isomorphism. Show that fl:H→G is also isomorphism.

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: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

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

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Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

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

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Q: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }…

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Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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Q: 1) Let G = {e,a,a',a²,a*,a,a°,a'] where a = e and let H = (a*) be a cyclic subgroup %3D %3D %3D of…

A: We have to find quotient group  G/H .

Q: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check  θ is homorphism θ is one-one…

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

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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: Prove or give counterexample. For any group G, Z(G) ≤ [G, G].

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

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

Q: = n and o(b) = m Let G be a group and suppose a, b E G satisfy ab = ba. Assume that o(a) : with (n,…

A: Given G is a group, such that a, b∈G satisfy ab=ba. Also, we have oa=n and ob=m with n, m=1.   To…

Q: 9*. Use Lagrange's theorem to prove: If g E G, G a finite group, then gG| = 1.

A: To prove the given statement, assume the following.

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: ation (1+xcos 2y)dr-x'sin 2 ydy = 0. %3D

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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: Theorem(7.9): If (H, *) is a subgroup of the group (G, *). then Va e G the pair (a+H a,+) is a…

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Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Let H = (a) be a cyclic group. Define f: Z → H by f(k) = a*. Prove that f is a surjective…

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Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: Let G be a group. If a G define C(a) = {g €G: gag¹=a}. Let a € G. Prove the following (a) Prove that…

A: The main objective is to prove the given statements.

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

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: 9.18 Suppose Hand K are subgroups of a group G and x, y € G. Prove that if Hx = Ky then H=K.

A: Given:suppose H and K are subgroups of a group G. and to prove:If Hx=Ky then H=K

Q: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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Q: Let w be a group and for each a Ew, let fa: Wow be defined as fa(x) = axa ¹ " Show that fa is an…

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