4. Let G be a group and g e G. Prove that the function f: G G given by f(x) = gx is a bijection.

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: . 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: Suppose thatf:G G such that f(x) and only if - axa. Then f is a group homomorphism if O a^2 = e a =…

A: See solution below

Q: 1. Let G be an abelian group with the identity element e. If H = {x²|x € G} and K = {x € G|x² = e},…

A:

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…

Q: (a) Let p: G → H be a group homomorphism. Show |p(x)| < |x| for all x E G.

A:

Q: Let G = {x ∈ R : x 6= −1} . Define △ on G by x△y = x + y + xy   Prove that (G, △) is an abelian…

A:

Q: Let G be an Abelian group and H = {x E G | |x| is odd}. Prove thatH is a subgroup of G.

A: Given: To prove H is a subgroup of G.

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: 3. Suppose that ged(m, n) = 1. Define f : Zn Z x Z, by f(r]mn) = ([T]m; [7]n). %3D (a) Prove that f…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Let f:G-G be a group homomorphism then H = {a € G:f(a) = a} is subgroup O True False

A:

Q: Prove if it is a group or not. 1. G = {x € R | 0 < x < 1},x * y = xy 1-x-y+2xy

A: *By Bartleby policy I have to solve only first one as these are all unrelated and very lengthy…

Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

Q: GX H G, X H. 19. Prove that a group Gis abelian if and only if the function f:G→ G given by f(x) =…

A:

Q: ii) Show that the function f (x) defined from the group (R, +) to the (R,×) by f (x) = e* is a…

A:

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: Exercise 8.6. Let G be a group. (a) Prove that G = {e} ≈ G. (b) Prove that G/{e} ≈ G. (c) Prove that…

A: 8.6 Let G be a group   (a) To Prove: G⊕e≅G (b) To Prove: G/e≅G (c) To Prove: G×e≅G

Q: 10. Let (G, *) be a group, and let H≤ G. Define N(H) = {x € G: x¹ *H* x = H} [Normalizer of H in G].…

A:

Q: Suppose thatf: G → G such that f(x) = axa². Then f is a group homomorphism if and only if ) a^2 = e…

A: Option C.

Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: 2) Let G be a group and H be a subgroup of G then : a) x • H = y • H + y•x=' € H. b) x • H = H → x e…

A:

Q: Suppose that f: G G such that f(x) = axa*. Then f is a group homomorphism if %3D and only if a = e O…

A:

Q: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

A:

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^2= e a =…

A: Since f is a group homomorphism , where f(x)=a∗x∗a−1,    x∈G. So a^-1=a implies self inverse implies…

Q: If f (x) is a cubic irreducible polynomial over Z3, prove that either xor 2x is a generator for the…

A:

Q: 2. Let G be a group. Show that Z(G) = NEG CG(x).

A: Let G be a group. We know Z(G) denotes the center of the group G, CG(x) denotes the centralizer of x…

Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

Q: If H≤G and let C(H) = {x element G| xh=hx for all h element H} prove that C(H) is a subgroup of G.

A:

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

A:

Q: Suppose that f:G G such that f(x) = axa'. Then f is a group homomorphism if and only if O a^2 = e O…

A: We will use property of homomorphism to solve the following question

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

Q: Suppose that f: G → G such that f(x) and only if = axa. Then f is a group homomorphism if a = e a^4…

A:

Q: Let G be an abelian group,fo f fixed positive integer n, let Gn={a£G/a=x^n for some x£G}.prove that…

A:

Q: Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = e O…

A: From the condition of group homomorphism we can solve this.

Q: Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = O a^4…

A:

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e a^3…

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

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

A:

Q: Q4: Consider the two group (Z, +) and (R- {0}, ), defined as follow if n EZ, f(n) ={1 if nE Z, %3D…

A: Homomorphism proof : Note Ze denotes even integers and Zo denotes odd integers.  So f(n) = 1 if n is…

Q: Let A be a group and let B be a group with identity e. Prove that (A x B)/(A x {e}) = B . Hint: Show…

A: Let A be a group and let B be a group with identity e. Let the operation in A is @ (say) and in B is…

Q: Let g be an element of a group G such that x 2 = g and x 5 = e. Then solve for x in terms of g.

A: Given : x² = g and x⁵ = e

Q: Let G be a group. Prove that Z(G) < G.

A:

Q: Prove if it is a group or not. 1. G = {x ≤R | 0 < x < 1},x * y = xy 1-x-y+2xy

A: *By Bartleby policy I have to solve only first one as these are all unrelated and very lengthy…

Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: Let G be a group. Let x, y e G be such that O(x) = 7, O(y) = 2, x^6 y = yx. Then O(xy) is O Infinity…

A:

Q: 4*. Let f G H be a group homomorphism. Prove: (a) If S G then f(S) 4 f(G) (b) Show by example that S…

A: To prove the stated properties of group homomorphisms

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

A: