Problem 4:(a) Compute the symmetry group of the pentagon and the hexagon. For each group, calculatethe cycle index.(b) A tetrahedron is a solid with four sides, each of which is an equilateral triangle.14Write down the permutations (in disjoint cycle notation) in S4 that correspond to rotationsof the tetrahedron in 3-dimensional space. How many are there? What is the order of each?What is the sign of each? What is the cycle index of this group?2.

Answered: n four sides,

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 6E: Let be the subgroup of . Find the distinct left cosets of in , write out their elements,...

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Problem 3. (1) Prove that (ac)(bd) = (abc)(abd) in Sn for distinct a, b, c, d ∈ {1, 2, 3, · · · , n}. (2) Prove that the product of every pair of transpositions in Sn can be expressed as a product of at most two cycles of length 3. (3) Prove that any permutation in the alternating group An with n ≥ 3 is a product of cycles of length 3.
Problem 4. Let G = A4. (1) List all of subgroups of G with order 2 or 3. (2) Explain why G has no subgroup of order 6. (3) Find the subgroup H of order 4. (4) List all left cosets of H in G.
Chapter 5, problem 8b Consider a cyclic group G and two positive integers A and B with the following two properties: (1). group G's order is A; (2) A divides B. Prove the number of elements in group G with order B is ϕ(B).
Let the cosets of the subgroup e in problem 3 be written as N=0+N and g+N, where g is 1. What is true about the order of the quotient group and the sum of this coset with itself (g+N)+(g+N) a. The order is 3, the product is g+N b. The order is 2, the product is g+N c. The order is 3, the product is N d. The order is 2, the product is N
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
Chapter 4 Problem 7: Denote G = (Q, +) as the group of rational numbers with addition. Answer the following questions: (1). Prove G is not a cyclic group. (2). Consider A, B as two nontrivial subgroups of G. Is A ∩ B also nontrivial? (3). Prove that for any a,b ∈ G, there exist h ∈ G such that a,b ∈ <x>.
Problem 7, Chapter 7, Section .3: Laurent Series Representation from the Complex Analysis for Mathematics and Engineering textbook, 5th Edition
Calculate the orders of the following a. The group of rotations of a regular tetrahedron (a solid with fourcongruent equilateral triangles as faces)b. The group of rotations of a regular octahedron (a solid with eightcongruent equilateral triangles as faces)c. The group of rotations of a regular dodecahedron (a solid with12 congruent regular pentagons as faces)d. The group of rotations of a regular icosahedron (a solid with20 congruent equilateral triangles as faces)
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Question 3 :Grapgh Theory --Discrete Math
Use the placebo group as group 1 and the Vytorin group as group 2. Need help with questions 1 through 4.
Discrete Math problem 1. (f) see attachment.

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