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:

Q: If you let G and H be two groups and consider the product group G x H and let K be the subset of G x…

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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…

A: Given that G be a group And For a∈G denote On the orbit of a under conjugate action. So SG=g∈G|ag=ga…

Q: Let G be a group and let a be an element of G. Prove that a func- tion fa : G → G defined by fa(g) =…

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

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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: 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…

Q: Let Φ: G to H be a group homomorphism such that Φ(g) ≠ eH for some non-identity element g in G,…

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

Q: Using the Third Isomorphism Theorem, or otherwise, prove that if H₁ and H₂ are subgroups of an…

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

Q: 3. Let G be a finite group of order |G| = n. Prove that G is isomorphic to a subgroup of Sym(G)…

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

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.…

Q: Number

A: Let, G be finite group and order of group G is n and n is finite. Let, G' be a group such that  ϕ: G…

Q: An automorphism of a group G is an isomorphism G→ G. (1) Prove that Aut(G), the set of all the…

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

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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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.

Q: Explain in detail if the following statement is true or false: " Suppose a group G acts on a set X;…

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

A: Let G,∘ and H,* be two group and ϕ:G→H be a homomorphism. Let ϕG is a non-empty subset of H. Since…

Q: Let : G H be a group homomorphism. Prove that Kery = {e} if and only if is injective.

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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…