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Unformatted text preview: Chapter 3 Di ff erential Forms 3.1 Exterior Algebra 3.1.1 Permutation Group • A group is a set G with an associative binary operation, · : G × G → G with identity, called the multiplication , such that each element has an inverse. That is, the following conditions are satisfied 1. for any three elements g , h , k ∈ G , the associativity law holds: ( gh ) k = g ( hk ); 2. there exists an identity element e ∈ G such that for any g ∈ G , ge = eg = g ; 3. each element g ∈ G has an inverse g 1 , such that g g 1 = g 1 g = e • Let X be a set. A transformation of the set X is a bijective map g : X → X . • The set of all transformations of a set X forms a group Aut( X ), with com position of maps as group multiplication. • Any subgroup of Aut( X ) is a transformation group of the set X . • The transformations of a finite set X are called permutations . • The group S p of permutations of the set Z n = { 1 , . . . , p } is called the sym metric group of order p ....
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 Spring '10
 Wong
 Algebra, Geometry, Multiplication, Permutation group

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