What is the center of a dihedral group Dn, when n is odd? {H,V} {Ro} {Ro, R180} {Ro, R90, R180, R270}

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

Q: Show that the cyclic group of n objects, Cn, may be represented by r", m = 0, 1, 2,...,n– 1. Here r…

A: Given  Cn is a cyclic group of n elements To show Cn is generated by some element r of Cn. That is…

Q: Consider the dihedral group D6.Find the subgroups of order 2,3,4 and 6

A:

Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: 7.)What is the center of a dihedral group Dn, when n is odd? {H,V} {Ro, R180} A В {Ro} {Ro, R90,…

A: We have to find

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

A:

Q: Let G be group, aeG ond lal=n. Show that laml = (m,n) %3D

A:

Q: Compute the left regular representation of the group G = {e,a, b} given by the group table below by…

A:

Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: 1. Construct the multiplication table for the the group Us = {1,a, a“, a°, a*} %3D 2mi where a = es.

A: As per the company rule, we are supposed to solve one problem, from a set of multiple problems.…

Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

Q: For a cyclic group of order 420, the number of generators are, (a) 66 (b) 76 (c) 86 (d) 96

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: Consider Lof order 4 of the dihedral group a normal subaroup H. Duz then prove that for each hC H.…

A:

Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is     a 1   b -1…

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: The cyclic group of order 12 acts on {1,2,..., 12} with the following cycle structure. (1)…

A: Given that,  the group of order 12 In this case of necklace, there is no difference between…

Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

Q: How many elements does the group of rotations of a regular hexagon have?

A: The group of rotations of a regular hexagon is D6. And as the order of the group Dn is 2n.

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: Write a Cayley table for the dihedral group D6 = {1, r,r^2,s,sr,sr^2|

A:

Q: Consider find Subgraup. Dihedral group D- of order 2,3,4 and 6.

A: A group G of two generators x and y of order n and 2 respectively with some relation is called the…

Q: Use the Cayley table of the dihedral group D3 to determine the left AND right cosets of H={R0,F}.…

A:

Q: Q2) If G = Z24 Group a) Is a G=Z24 cyclic? Why b) Find all subgroups of G = Z24 c) Find U,(24)

A: Given that G=ℤ24. a) Then G is generated by the element 1. That is, 1=1,2,3...,22,23,0=ℤ24.…

Q: is a group with identity (eg, eH).

A:

Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

A:

Q: The dihedral group associated with the symmetries of a square has elements. O a. 16 О Б. 2 О с. 4 O…

A: Concept: A dihedral group is the group of symmetries of a regular polygon, which includes rotations…

Q: Show that (ℤ,∗)?ℎ??? r ∗s = (r +s)−(r ∙ s)??? ??? r,s ∈ ℤ is group using variable r, s and t.

A:

Q: 2. Consider the group G of symmetries of a regular pentagon, generated by a rotation a and a…

A:

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

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

A:

Q: The group ((123)) is normal in the symmetry group S3 and alternating group A4.

A:

Q: The group U(15) is an internal direct product of the cyclic subgroups generated by 7 by 11, U(15) =…

A: We  have to check that U(15)= <7>×<11>  Or not. Concept: If  n =p1.p2 Where p1 and p2…

Q: (d) Find the cosets of the quotient group (5)/(10), and determine its order.

A:

Q: List all the elements of the cyclic subgroup of U(15) generated by 8. 2. Which of the following…

A: We have to find the all elements of cyclic subgroup of U(15) generated by 8.

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

Q: Consider a dihedral group Dy= (a,b:a'- *. (ab)`=t). (i) Find all the conjugacy classes of Dy Give…

A:

Q: In D4, the centralizer of the group at H is equal to?

A:

Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A:

Q: Let X = Cay(Z12, {±2, 6}) and Y = Cay(D12, {P, p°, T}), where D12 is dihedral group of order 12,…

A: Cayley graph or Cayley diagram is used to study the abstract nature of a group by drawing graphs…

Q: List the elements of the quotient groups of (a) (4Z, +) in (Z, +) (b) Z30/(6) (c) Z30/(J/H), where J…

A: Quotient group GH ={ Ha | a∈ G} where H is normal subgroup of group G. Here al given groups are…

Q: Show that the group of rotations in R³ of a 3-prism (that is, a prism with equilateral ends, as in…

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12:

Q: Which of the following is cyclic group а. R b. Z С. Q d. C

A:

Q: if it was ifit S={a+b/2 :a,beZ}and (S,.) where(.) is a ordinary muliplication prove that his group?

A:

Q: For G = (Z5 ,+s) , how many generators of the cyclic group G? 5.a O 1.b O 3.c O 4.d O 2.e O

A:

Q: Which of the following is group under multiplication? (A) Q (B) Q-{0} (C) Q-{1} (D) Q-{0,1}

A:

Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…

A:

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.