True or False. If true, explain. If false, give a counterexample. (f) Let ϕ : G → G' be a group homomorphism with |G| = 12, then the image of G, ϕ[G] may have order 9. (g) A map φ : G → H that is one-to-one and onto is a permutation. (j) In a group G, and g ∈ G. If |g| = p where p is a prime, then |G| = p^k for an integer k.

Q: . Let p: G → H be a group homomorphism. a) p(1g) = 1µ; b) p(a-1) = [9(a)]-1,va E G;

A: We have: to use definition of homomorphism:

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Q: Let G be a group and let G act on itself by conjugation. For a e G, denote Oa the orbit of a under…

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

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Q: Theorem Let f: G H be a group homomorphism. Then, Im f≤ H.

A: Let us consider the mapping f:G→H . Then f is group homomorphism if  f(x·y)=f(x)·f(y) where, x,y∈G.…

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Q: Consider the map o : G¡ → G2 defined by: 9(a) = a-! %3D (a) Does o define group homomorphism? (b)…

A: Property of homomorphism of groups....

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Q: Suppose that f: XY is a homomorphism (of either groups or rings). Prove that f is injective if and…

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Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

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Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

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

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Q: Let G and H be finite groups. Prove the following statements. (a) If a: G H is a homomorphism, then…

A: Given that  and  are finite groups.We need to prove the following.(a) If  is a homomorphism, then…

Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

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A: a) Let G and H be finite groups andϕ : G → H a homomorphism. Then|ϕ(G)| · |Ker (ϕ)| = |G|  (By…

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A: The objective of this question is to prove the relationship between the orders of two subgroups H1…

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A: The objective is to prove that G is isomorphic to a subgroup of symmetric group.

Q: et f be an isomorphism from a grou onto a Group H. et a be an element in G and the orde Fa is 5. it…

A: We have to choose the correct answer from the given option according to the given condition

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Q: a) Check whether G = (3n, ·) is a group, where n ∈ Z. b) Let p1 (x) = x3 + 1 and p2 (x) = x4 + x.…

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Q: Exercise 3.1. Let G be a group. Given g € G, define ig : G → G by ig(x)=gxg, Vx Є G. (1) Show that…

A: Let  G  be a group and g∈G1. Show that the map , defined by  for all , is an automorphism of  G .2.…

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A: As per the policy, we are solving 3 subparts of the given question. Please repost the question and…

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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. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

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Q: et G = {fab | a, b € R, a 0}, where f(x) = ax + b for all real x. Consider the group G under the…

A: Given that G is a group defined as            G = { fa,b | a, b ∈ R , a≠ 0} .   where   fa,b(x)=ax+b…

Q: Let : G → H be a group homomorphism. Show that is a subgroup of H. Im(6) := {p(g) | g = G}

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Q: Prove that if G is a cyclic group and f G →→ H is a homomorphism, then Im(f) is a cyclic subgroup of…

A: Given that .So let We have to show that  is cyclic subgroup of H .Subgroup test :Let,Now…