Q) Let H be a subgroup of (R.+), let B= (e" :aeH). A) Prove that B is a subgroup of (R,..) B) Is B normal in G? (Explain your answer)

Q: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Let G = Z4 x Zp, and S = {(1,0), (3,0), (2,1), (0,1). Prove that Cay(G, S) is not normal Cayley…

A: To prove cay(G , S) is  a normal Cayley graph you need to prove that for any a∈G a-1Sa=S ∀a∈Z4×Z2=G…

Q: Let ş #hti#j,+k? be a group f querterniong, ptove That then q''s nilpotent.

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

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Q: 2) Let H be a normal subgroup of G. If| H|-2. Prove that H is contained in the center Z(G) of G.

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Q: Let H be the subgroup (10) of Z15. (i) What's the order of H? (ii) What's the number of left cosets…

A: (1) Let G be a cyclic group generated by 'a'.G = <a> = {ai : iEZ}If |G| = |a| = nthen order of…

Q: Compute the orthaganal piojectian of ? onto the lina L through -6 and the origin

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Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

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Q: I. Provide a two-column proof to the following statements. 1. Prove Theorem 2: Center is a subgroup…

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

Q: a. Prove or Disprove. If H is an abelian normal subgroup of G then H be contained in Z(G).

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Q: Ql: Prove that (Q\{0},x) is a subgroup of (R\{0},x).

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Q: 4) Use Green's Theorem to evaluate f √1+x³dx + 2xydy. C is the triangle with vertices (0,0), (1,0),…

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Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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Q: (8) Show that the map f(z) = 1/z is a bijective holomorphic map from S? to itself, using the charts…

A: Note: A function is holomorphic if it is analytic everywhere except one point. Given: fz=1z, z≠0

Q: Suppose that Find the Jacobian Metrix for map.

A: Given y1=x1x2-x2x4y2=x12-x42y3=x1x2x3x4

Q: Let H & K be tavo to "HA'G, then Arnormal suograup subgroups group G. 4 Prove that HA K AG

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Q: ⊙O and ⊙P are given with centers (−2, 7) and (12, −1) and radii of lengths 5 and 12, respectively.…

A: Two circles ⊙O and ⊙P are given with centers (-2,7) and (12,-1) and radii of lengths 5 and 12…

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: Let H be the subgroup of all rotations in Dn and let Φ be an automorphismof Dn. Prove that Φ(H) = H.…

A: Given: The subgroup is H The automorphism of Dn  is ϕ To prove that ϕ(H) is H. Let the subgroup H of…

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: Exhibit a plane figure whose plane symmetry group is Z5.

A: A pentagon can be modified such that the only symmetries are given by rotation.

Q: Coxeter

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Q: Let H = where pis a rotation by an angle 360° in Dn. Then, H is a n subgroup of Dn but not normal…

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Q: be a Galois extension with Galois group G. Then F = KG.

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Q: let FCK be a Galois extensions wi Gates group by. let X = (HCW / 11² of cy! and y = [/cp| this…

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Q: 72 please

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Q: Exercise 12, We identify S, with the sulgreup { 1,2,3} f Sy. Denode by v the four-subgroup (a) Prove…

A: We shall solve first three question only as per company guidelines. For others kindly post again by…

Q: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: Prove that a normal subgroup need not to be a characteristic subgroup.

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

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Q: 4. What is the image of the half strips as shown on the figure, under the mapping z iz? Under the…

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Q: Let G be the group of symmetries of a circle and R be a rotation of the circle of V2 degrees. What…

A: Given:

Q: Let H be a subgroup of G. Show that if aH Deduce that H is normal in G if and only if every left…

A: Let's first show abH⊆Hab Let,  abh=abh=ah1,b Since Hb=bHfor some h1∈H Therefore,  abh=ah1b  Since,…

Q: b' e GL(2, IR) а Is Ga subgroup of GL(2, IR)? Let G

A: Note that, the general linear group is

Q: 4. Select the sequence of transformations that maps triangle LAP onto triangle L'A'P'. Ay P'

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Q: Find a non-trivial, proper normal subgroup of the dihedral group Dn-

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Q: -{8 a|a, b, d ER, ad # 0 }. Is Ha normal subgroup of GL(2, R)? Let H

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Q: Can Z121 have a subgroup of order 20? Explain.

A: Subgroup of order 20

Q: A transformation that is a composition of isometries is also an isometry. O True False

A: A transformation that is composition of isometries is also an isometry - Let,    (X1, d1), (X2, d2),…

Q: a dihedral group Dyz (a,bia'- b°- (ab)`=4>. Consider i) Discuss all the possible Commutators of Dy…

A: Let us first find the conjugacy classes for D4.

Q: -) Show that QISn] is Galois over Q with Galois group isomorphic to (Z/nZ)*.

A: Let, H⊂Galℚξp/ℚ be a subgroup  defining αH∈ℚξp to be, αH=∑α∈Hσξp One can express the fixed subfield…

Q: think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal…

A: A subgroup H of h is called normal subgroup of h if θH⊆H  ∀θ∈AutG

Q: Q) Let H be a subgroup of (R.+), let B= (e:ae#). A) Prove that B is a subgroup of (R'..) B) Is B…

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Q: The following is a subgroup of GL2(R) (which you do not need to prove): a H = ad Consider 2 1 a = b…

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Q: Let H be a subgroup of G. Show that if aH = Deduce that H is normal in G if and only if every left…

A: Given:- Let H be a subgroup of G. To prove:- If aH=Hb for some a,b ∈G then aH=Ha. also if H is…

Q: Exercise 3. Part A: Let G = Sn and let H = {a € G|a(1) = 1}. Show that H is a subgroup of G.

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Q: 3. Show that a reflection a in the x-axis defined by o(x, y) = (x, -y) is an isometry.

A: The solution of given problem is followed by

Q: to be the subset Let H be a subgroup of G. Define the normalizer of H in NG(H) = {g e G| gHg H}. (i)…

A: Let H be a subgroup of G. The normalizer of H in G is given by NGH= g∈G  gHg-1=H  A nonempty subset…

Q: 6. Show that the mapping given below is (a) one to one (b) is not an isometry.

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.