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Foundations of Quantum Gravity by James Lindesay

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Appendix A Addendum for Chapter 1

A.1 Groups and special relativity

A.1.1 Fundamentals of group theory

Properties of groups

A group is a set of elements that have the following properties:

  • Contains the identity transformation 1;
  • If E and E′ are elements in the group, then there exists a group operation (generically called “multiplication”) that always produces an element of the group, E″ ≡ E′ · E (closure);
  • For every transformation element E, there exists in the group an inverse element E1, where E1· E = 1;
  • The group operation is associative, E″ · (E′ · E) = (E″ · E′)· E.

A particular type of group satisfies an additional property. For an abelian group, the group operation yields the same answer regardless of the order, E′ · E = E · E′. ...

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