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Properties of the Octahedral Group

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a) Determine the class equation of the octahedral group.
b) This group contains two proper normal subgroups. Find them, show that they are normal, and show that there are no others.

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Solution Preview

a) The octahedral group O has the following six conjugacy classes:

C1: The identity
C2: The six 90-degree rotations
C3: The eight 120-degree rotations
C4: The three 180-degree rotations about a four-fold axis
C5: The six 180-degree rotations about a two-fold axis
C6: The 24 reflections.

Therefore the class equation of O is

48 = 1 + 6 + 8 + 3 + 6 + 24.

b) The proper normal subgroups of O are the subgroup ...

Solution Summary

We derive the class equation for the octahedral group and determine its normal subgroups.