*10*

*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 *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*^{−1}*g* = *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 ...

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