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Physical Mathematics by Kevin Cahill

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10

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Group theory

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 fG and gG, the product fgG (closure);

2   if f, g, and h are in G, then f (gh) = (fg)h (associativity);

3   there is an identity eG such that if gG, then ge = eg = g;

4   every gG has an inverse g−1G such that gg−1 = g−1g = e.

Physical transformations naturally form groups. The product TT represents the transformation T followed by the transformation T′. And both T″ (TT) and (TT′) T represent the transformation T followed by the transformation T′ and then by T″. So transformations ...

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