16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

Q: let G be an abelian group. And let H = {r :z€ G) show that H < G? %3D

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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: Define on S = Q × Q* the operation (a, b) * (x, y) = (ay+ x, by). Recall that Q* = Q\{0}. 1. Explain…

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Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: 10. Let E = Q(V2, V5). What is the order of the group Gal(E/Q)? What is the order of Gal(Q(V10/Q)?

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Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

A:

Q: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

A:

Q: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

A:

Q: Define on R the operation * by x*y = X+y+k, for all x,y element of R and k is fixed real number. The…

A: We have to check

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

A:

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

A:

Q: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

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: c) Show that Z,,+, is a cyclic group generated by 3

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?

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

A:

Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: 3. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

A:

Q: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

A:

Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

A:

Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are…

A: Given:  Let G be a group. ZG be its center.  We know that ZG=z∈G: ∀g∈G,zg=gz ....i First we will…

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: 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: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

A:

Q: Show that if aEG, where G is a group and |a| = n then : %3D a' = a' if and only if n divides. -j

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Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

A:

Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Define * on Q by a +b= qb Is Q a group under *? 210

A:

Q: 4) Let G. be Graup and aE G La> ç Cala)? give Is Prove OY Counter example G. H, k Such (2) Let be…

A: Centralizer of 'a' in G- Let a be a fixed element in a group G. Then the centralizer of 'a' in G is…

Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: Find the group homomorphism between (Z, +) and (R- (0},.)

A:

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

A:

Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

A:

Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.