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

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Advance Math

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 groupsand their elements. After clearly naming the elements in some way,provide tables for each group. Describe them as a group ofpermutations on the vertices.

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

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

d. Next, carefully describe all the subgroups of each of thesegroups. Be sure to provide information about the structure of eachsubgroup, their order, the order of their elements. Providegenerator(s) where possible.

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

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