o= (1,8,5,7)(2,4) y= (1,3,2,5,8,4,7,6) a. Compute oy. b. Determine whether oy is odd or even. 2.2 Let f: → be defined by f(x) = 3x - 3. Prove or disprove that f is an isomorphism from the additive group onto itself.

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: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Show that the frieze group F6 is isomorphic to Z ⨁ Z2

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Q: Define ∗ on Z by x ∗ y = x + y + 4. Then: (d) Show that every element of Z is invertible with…

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Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

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

A: Given:  x '= x ,y' = cx + dy d

Q: 2.2 Let f: → be defined by f(x) = 3x - 3. Prove or disprove that f is an isomorphism from the…

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Q: Define the mapping 7: R²→R by π((x,y))=x. (Note that R is a group under addition with identity 0).…

A: Here we use the definitions of group homomorphism and the kernel of it . Which are given in solution…

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....

Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k E U(n) such that k2…

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

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Q: Let G = (Z;, x,) be a group then the order of the subgroup of G generated by 2 is О а. 6 O b. 3 О с.…

A: We have to find order of subgroup of G generated by 2.

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

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Q: In the following problems, let G be an abelian group and prove that the set H described is a…

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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…

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Q: 3. Let f : (R\{0},-) (R\{0}, -) be group homomorphism defined by f(a) = |a|. Then ker(f) %3D =...

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Q: Let f. → (R, +> be defined by f(x) = 3x- 3 Prove or disprove that f is an isomorphism from the…

A: This is related to isomorphism

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: LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

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Q: In the set of real numbers R there is an operation defined as: x x y = Vx³+y³ prove that (R, x) is…

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Q: True or false? The group S3 under function composition ◦ is not a cyclic group

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Q: Let f be the homomorphism f Z1s → GL(2) of groups such that (1) = A, where A = a) Give the…

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Q: Consider the group G (x E R]x 1} under the binary operation : ** y = xy-x-y +2 If x E G, then x =…

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Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. Prove that…

A:

Q: 2. Consider the groups (R, +) and (R x R, +). Define the map : RX (RX R) → Rx R defined by r(x, y) =…

A: Given: The group is, (ℝ, +) and (ℝ×ℝ, +) The map is, ℝ×(ℝ×ℝ)→ℝ×ℝ To find: a) The defined map is an…

Q: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

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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: Let S= \ {-1} and define an operation on S by a*b = a + b + ab. Prove that (S,*) is an abelian…

A: Given: The operation on S=R\-1 is defined by a*b=a+b+ab To prove: That (S,*) is an abelian group.

Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

A:

Q: 5. Consider the group (R+, 0). Prove that the function F: R -R given by: F (x.y) = (x +y.r-y) is a…

A: F: R2 → R2F (x, y) = (x +y, x-y)Let (x1, y1) ,  (x2, y2) = R2 (x1, y1) +  (x2, y2) =  (x1+ x2, y1+…

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

A: Given that f from G to G is a function defined by f(x)=axa2 Then we need to find a necessary and…

Q: 3. Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the…

A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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

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Q: Let ?: ℤ × ℤ → ℝ∗ be defined by p((a, b)) = 2a 3b   (i) Prove that p is a group homomorphism.…

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Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

A: The set G with binary operation * is said to form a group if it satisfies the following properties.…

Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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

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Q: 25. Prove that R* x R is a group under the operation defined by (a, b) * (c, d) = (ac, be + d).

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Q: Let x, y be elements in a group G. Prove that x^(−1). y^n. x = (x^(−1).yx)^n for all n ∈ Z.

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Q: For the homomorphism : D4 → ({1, −1}, •) defined by Þ(a) = { 1 -1 if a is a rotation if a is a…

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Q: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

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

A: A mapping f from a group (A,.)  to a group (B,*) is called a group homomorphism if f preserves the…

Q: Suppose that fG G such that f(x) = axa. Then fis a group homomorphism if and only if O a^3 = e a^2 e…

A:

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

A: