1. Let G be a p-group acting on a finite set X. Let XG be the set of fixed points of this action: XG = {x € X, such that g.x = x for any g € G}. Show that #XG = #X mod p. (in other words, show that p divides the difference #X − #XG).

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…

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Q: 9.12 Let G be a group and for elements a, b e G let a R b mean that there exists an element x = G…

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Q: Let G:- [0, 1) be the set of real numbers x with 0<x<1. Define an operation + on G by ** y={x+y if…

A: Solution:-  Given   G=0,1   be the set of real numbers xwith  0≤x&lt;1.  The operator ∗on G is given…

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Q: 5. Let a E G be fixed. Define y: G → G by p(x) = axa-1 for all x E G. Prove that p is a…

A: To find - Let a∈G be fixed. Define ψ:G→G by ψx=axa-1 for all x∈G. Prove that ψ is a homomorphism.

Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: 7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E…

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Q: 2. Let a € R and n € Z0. Show that of the numbers a, 2a, distance from an integer is at most 1/n.…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Exercise 3. Let T be a set such that |T| ≥ 3 and let G = Sym(T) be the group of permutations of T.…

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Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

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Q: 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…

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Q: 1. Let A be a non-empty set and (G, .) be a group. Let F be the set of all functions from A to G.…

A: Let A be a non empty set. For f,g∈F with the operation f*gx=fx.gx To prove that F,* is a group.…

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Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: unctions are well-defined group homomorphisms? Se elections; you cannot score less than 0 on this…

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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…

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Q: 4. For any sets A, B and any function f: A → B, we define the preimage of a subset SCB to be the set…

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Q: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

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Q: Show that if g: R->R' is a homomorphism, then Ker(g) = { 0 } if and only if g is one to one

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Q: (B) Let G be a set of all real numbers except 0. Define on G by a b = |a|b, where lal is the…

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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…

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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 x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: It is given that x and y be two vertices of a Cayley digraph. We can write two paths as a1,a2,a3,…an…

Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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Q: 1. Let (G, +) and (H,*) be groups. The direct product of G and H is the set G × H equipped with…

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Q: 2 20. Let G be a group with subgroup H. Define a relation on G as follows: a b if b-¹a EH. Prove…

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Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

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Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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Q: If f:G\rightarrow{}G' is an isomorphism of groups, show   that the order of an element a∈G is equal…

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Q: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

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Q: ACR group A = {x+y: −1 < x≤0, 2 < y ≤ 3}. Define group A's infinum (inf A).

A: As  -1 &lt; x And  2 &lt; y Then   -1+2 &lt; x+y I. e. 1 &lt; x+y

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Q: (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove…

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Q: Let f.g:(G,.) (G')and h(m) f(m). g(m), Vm € G, them h is group homomorphism if f is homomorphism O f…

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Q: Exercise 14. Let G₁ and G₂ be a groups. Then the set G₁ X G₂ is a group with the following binary…

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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: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________