Ideas and Terminology

Inverse of g
The function which makes the composition with g be the identity function. For example, the inverse of rotation by Pi/2 is rotation by 3 Pi/2.

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.

Further Reading for Undergraduates

Geometry of Surfaces by John Stillwell
This book is a very good reference for geometric intuition, groups, fundamental domains and quotient spaces. This book only looks at actions which have no fixed points, so my examples will not be there. (You will have to see my Ph.D. thesis for those!)

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.