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Generalities refers to the definition of basic concepts. The goal here is to define group, field, vector space and algebra. The chapter starts with the definitions of associativity, identity, inverse and closure and use these to define group. This is followed by the definition of subgroup and invariant subgroup. The cyclic group is introduced to provide an example of a finite group. Next, homomorphism, isomorphism, realization and representation are defined. A canonical description of an *N*-dimensional vector space is given in terms of *N*x1 column matrices. In conclusion, the commutator of two matrices is defined and the Jacobi identity is presented. Biographical notes on Galois, Abel and Jacobi are given.

*Keywords: *
group;
field;
vector space;
algebra;
isomorphism;
representation;
Jacobi identity

*Chapter.*
*2514 words.*

*Subjects: *
Mathematical and Statistical Physics

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