Action or Group Generated by g The set of all compositions of combinations of g and the inverse of g. For example, the group generated by rotation by Pi/2 is the set of 4 elements containing rotations by Pi/2, 2 Pi/2, 3 Pi/2, 4 Pi/2.
Quotient Space of the Basketball by G Two points in the new space are the same if you can get from one to another on the basketball by some g in G. To see this geometrically, we find a fundamental domain (see below), and sew up the edges. For example, a football in Example 1 on the other side of this sheet.
Fundamental Domain on the Basketball Corresponding to an Action G A largest region or wedge on the globe for which any g from the action moves a point inside the region to a point outside of it. For example, an orange peel wedge as in Example 1 on the other side of this sheet.
Algebra by Michael Artin See Chapter 5 - Symmetry. This is a good reference for the finite groups arising as the symmetries of the platonic solids. For example, the icosahedral group is discussed here. See especially pages 164 and 184.