Group of Symmetries of a Cube a. Carefully describe the group of symmetries of a cube....

Group of Symmetries of a Cube

a. Carefully describe the group of symmetries of a cube. Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices.

b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices.

c. What are the POSSIBLE orders for any subgroups of each group? Explain.

d. Next, carefully describe all the subgroups of each of these groups. Be sure to provide information about the structure of each subgroup, their order, the order of their elements. Provide generator(s) where possible.

e. Answer these questions about each group described in part a making sure to give reasons: Are any of these groups cyclic? Are any of these groups abelian? Which groups are cyclic? Which groups are abelian? Are there subgroups of every possible order? Which subgroups (in each group among the different groups) are isomorphic? How do you know they are isomorphic or not?

Expert Solution

Colby Messinger answered 1 year ago

Group of Symmetries of a Rectangle a. Carefully describe the group of symmetries of a rectangle...

Group of Symmetries of a Rectangle a. Carefully describe the group of symmetries of a rectangle Describe the types, the orders, and the structures of the groups and their elements. After clearly naming the elements in some way, provide tables for each group. Describe them as a group of permutations on the vertices. b. Next, carefully describe each of these groups as subgroups of some permutation group. Be sure to provide reasons for your choices. c. What are the POSSIBLE...

Find all the subgroups of the group of symmetries of a cube. Show all steps. Hint:...

Find all the subgroups of the group of symmetries of a cube. Show all steps. Hint: Label the diagonals as 1, 2, 3, and 4 then consider the rotations to get the subgroups.

Find the symmetries of the Platonic Solids: Cube, Tetrahedron, Octahedron, Icosahedron, and Dodecahedron. I. You need...

Find the symmetries of the Platonic Solids: Cube, Tetrahedron, Octahedron, Icosahedron, and Dodecahedron. I. You need to describe with words the planes of reflectional symmetry and identify how many there are in any of the categories found. II You need to describe with words the axes(lines) of rotational symmetry. For each type of axis, determine how many there are and the order of rotation.

Let D8 be the group of symmetries of the square. (a) Show that D8 can be...

Let D8 be the group of symmetries of the square. (a) Show that D8 can be generated by the rotation through 90◦ and any one of the four reflections. (b) Show that D8 can be generated by two reflections. (c) Is it true that any choice of a pair of (distinct) reflections is a generating set of D8? Note: What is mainly required here is patience. The first important step is to set up your notation in a clear way,...

Consider the subgroup of S(4) and the group of symmetries of the rectangle discussed after Theorem...

Consider the subgroup of S(4) and the group of symmetries of the rectangle discussed after Theorem 5.3.1. There we defined one isomorphism between these groups but this is not the only one. Find all the rest. The isomorphism defined in 5.3.1 is the one between S(4) and the symmetries of a rectangle.

Find two distinct subgroups of order 2 of the group D3 of symmetries of an equilateral...

Find two distinct subgroups of order 2 of the group D3 of symmetries of an equilateral triangle. Explain why this fact alone shows that D3 is not a cynic group.

Find all of the symmetries of each of the following figures in E2 (Euclid Plane). Describe...

Find all of the symmetries of each of the following figures in E2 (Euclid Plane). Describe each symmetry precisely, e.g., explain which line you are reflecting over, what point you are rotating about, by how much, etc. A point A Two (distinct) points A and B. Two (distinct) lines l and m intersecting in a point P. (Warning: Your answer may depend on the angle between l and m.)

Describe the strategy that will be used to manage stakeholders for the Water Cube project. Describe...

Describe the strategy that will be used to manage stakeholders for the Water Cube project. Describe the purpose of a stakeholder management strategy. Describe the methods that will be used to identify project stakeholders. Describe the methods that will be used to determine stakeholder communications requirements.

Describe Menaechmus’s method of duplication of the cube, using parabolas

- The cube length is 3.23 cm. What is the volume of the cube? _ ml...

- The cube length is 3.23 cm. What is the volume of the cube? _ ml - How many significant figures should be in the calculated volume? - The measured mass of the cube is 43.808 g. Using that number combined with whatever the volume is, determine the cube's density (in g/mL) ? - Based on the density that was calculated, which type of plastic was used to make the cube? Low Density Polyethylene, High Density Polethylene, Nylon, Polystyrene, PVC,...

Question