Reference: Abstract Algebra, Third Edition, I N Herstein
First consider the following: Let S be the plane, that is, and consider defined by and ; f is the reflection about the y-axis and g is the rotation through 90 degrees in a counterclockwise direction about the origin. We then define , and let * in G be the product of elements in A(S). Clearly, identity mapping;
It is a good exercise to verify that and G is a non-abelian group of order 8. This group is called the dihedral group of order 8. [Try to find a formula for that expresses a, b in terms of i, j, s and t.
II) Let S be as in above example and f the mapping in above example. Let and let h be the rotation of the plane about the origin through an angle of in the counterclockwise direction. We then define and define the product * in G via the usual product of mappings. One can verify that identity mapping and . These relations allow us to show that (with some effort) G is a non-abelian group of order 2n. G is called the dihedral group of order 2n.