10.1 What is a group?
A group G is a set of objects f, g, h, . . . and an operation called multiplication such that:
1 if f ∈ G and g ∈ G, the product fg ∈ G (closure);
2 if f, g, and h are in G, then f (gh) = (fg)h (associativity);
3 there is an identity e ∈ G such that if g ∈ G, then ge = eg = g;
4 every g ∈ G has an inverse g−1 ∈ G such that gg−1 = g−1g = e.
Physical transformations naturally form groups. The product T′ T represents the transformation T followed by the transformation T′. And both T″ (T′ T) and (T″ T′) T represent the transformation T followed by the transformation T′ and then by T″. So transformations ...