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.

Expert Solution

Colby Messinger answered 2 years ago

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...

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.

SupposeG=〈a〉is a cyclic group of order 12. Find all of the proper subgroups of G, and...

SupposeG=〈a〉is a cyclic group of order 12. Find all of the proper subgroups of G, and list their elements. Find all the generators of each subgroup. Explain your reasoning.

Consider the group Z/24Z. (a) Find the subgroups 〈21〉 and 〈10〉. (b) Find all generators for...

Consider the group Z/24Z. (a) Find the subgroups 〈21〉 and 〈10〉. (b) Find all generators for the subgroup 〈21〉 ∩ 〈10〉. (c) In general, what is a generator for 〈a〉 ∩ 〈b〉 in Z/nZ? Prove your assertion.

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...

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,...

(a) Show that a group that has only a finite number of subgroups must be a...

(a) Show that a group that has only a finite number of subgroups must be a finite group. (b) Let G be a group that has exactly one nontrivial, proper subgroup. Show that G must be isomorphic to Zp2 for some prime number p. (Hint: use part (a) to conclude that G is finite. Let H be the one nontrivial, proper subgroup of G. Start by showing that G and hence H must be cyclic.)

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.

(1) Let G be a group and H, K be subgroups of G. (a) Show that...

(1) Let G be a group and H, K be subgroups of G. (a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a subgroup of G. (b) Show that if H and K are both normal subgroups, then HK is also a normal subgroup. (c) Give an example of subgroups H and K such that HK is not a subgroup of G.

PLEASE SHOW ALL STEPS WITH EXPLANATIONS Let G be a group (not necessarily an Abelian group)...

***PLEASE SHOW ALL STEPS WITH EXPLANATIONS*** Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5

Question