Problem 2. Let f be a homomorphism from a group G into a group H. Prove that f is one to one if and only if ker f = {e}.

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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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: 2. Let G be a group and H1, H2 <G subgroups. (a) Suppose |H1| = 12 and |H2 = 28, prowe H1 n H2 is…

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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: Theorem 3.34: Let (G, *) be a group, then (1) {e}AG. (2) GAG (3) Cent(G) AG. Proof: (H.W.)

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Q: Prove or disproe: If C is a groun and a hsC then (ab)-1 h-1a-1

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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: 5. Let G be a group. (a) Prove that Inn(G) is a subgroup of Aut (G) (b) Prove that for any € Aut(G)…

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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: 2. Let T = {z E C : |z| = 1} be the multiplicative group of unit circle. Let A be a nontrivial…

A: Given: Group T is a multiplicative group of unit circle. Group A is a non-trivial abelian group and…

Q: Problem 1. Let f be a homomorphism from a group G into a group H. (a) Prove that ker f sG (the…

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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: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

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: 3. Define Lie group.

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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 groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G)…

A: Let us prove above results

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

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Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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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: set P(S)

A: According to the question, The power set P(S) of a set S is the set of all subsets so, of S,…

Q: (a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the…

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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: 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: (G, .> be a group such that a.a = e for all a E G. Show that G is an abelian grou 2. Let

A: We have to solve given problem:

Q: 4. Let G and G be groups and let y be an isomorphism from G to G. Prove that if KCG, then y(K) CG.

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Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

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: Problem 3. Prove that if m ± 1n, then the symmetric groups S and S. are not isomorphic.

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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: 1. Let G be the set of integers under addition and H = {-1,1} be a multi- plicative group. Show that…

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Q: 10. The set of automorphisms of a group is a group under the operation of function composition. True…

A: Isomorphism : A bijective mapping  that preserves binary compositions in both groups is called…

Q: 3. Let G be a group with identity e. Show that if a ★a=e for all a in G, then G is Abelian.

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