be a function from G to Sx, and for g E G and x E X, let g · x = a(g)(x). Prove that this defines a group action if and only if the function a is a group homomorphism.

Q: subgroups of G. Let x,y e G. We define the relation x ~ y by the following definition: x ~ y iff y =…

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Q: Prove that the following functions are homomorphisms. Determine the kernel of each function. - a) f:…

A: To prove that a function is a homomorphism, we need to show that it preserves the group operation.…

Q: 15. Let G be a group and let g be in G. Define a function :G → G by ,(x) = gxg Show that :G→G is…

A: Let G be a group and let g be in G . Define a function ϕg : G→G by ϕgx=gxg-1. To prove: ϕg : G→G is…

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: Prove that the mapping from U(16) to itself given by x → x3 is anautomorphism.

A: To show that the mapping from U16 to itself given by x→x3 is an automorphism, we need to show that…

Q: Let Φ be an automorphism of C*, the group of nonzero complexnumbers under multiplcation. Determine…

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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: Determine which of the following are homomorphisms. Justify your answers. Recall that R denotes the…

A: Here given that two groups ,i) Given a function So for .So  is homomorphism .ii) Given function is…

Q: 1*. Let f G H be a group homomorphism. Prove: (a) If l is the identity of G then /(lg) is the…

A: To prove the required properties of a group homomorphism

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: How can I prove that f is one-to-one (injective)?

A: We know, a function/map f is one to one (injective) iff          For a ≠ b, f(a) ≠ f(b) for any two…

Q: A group G acts faithfully on a set S if g * s = s for all s € S implies g = e. Show that G acts…

A: Proof 1: If G acts faithfully on S then mg≠mg', where g≠g' ∈G. Let G acts faithfully on S. Assume to…

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: 1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6:…

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Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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Q: 1. Let 0(V) be the set of all orthogonal transformations on V. Prov O) is a group with respect to…

A: Let O(V) be the set of all orthogonal transformations on V. The determinant of an orthogonal matrix…

Q: If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism,…

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Q: Show that the determinant gives a group homomorphism det: GL(n,R)->R*, where R*=R-{0} under…

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Q: Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

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Q: 5. For each a in the group G, define a mapping ta: G→ G by ta(x) = axat. Prove that to is an…

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Q: 2. For an arbitrary set A, the power set P(A) = {X | X C A}, and addition in P(A) defined by X+ Y =…

A: Recall the following. Definition of group: Suppose the binary operation * is defined for element of…