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 2 years ago

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