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differential geometry w notes from teacher_Part_82

differential geometry w notes from teacher_Part_82 -...

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Chapter 7 Homology Theory 7.1 Algebraic Preliminaries 7.1.1 Groups A group is a set G with a binary operation, · , called the group multiplication, that is, 1. associative, 2. has an identity element, 3. every element has an inverse. A group is Abelian if the group operation is commutative. For an Abelian group the group operation is called addition and is denoted by + . The identity element is called zero and denoted by 0. The inverse element of an element g G is denoted by ( - g ). Let G and E be Abelian groups. A map F : G E is called a homomor- phism if for any g , g G , F ( g + G g ) = F ( g ) + E F ( g ) , where + G and + E are the group operations in G and E respectively. In particular, F (0 G ) = 0 E , and F ( - g ) = - F ( g ) . 163
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164 CHAPTER 7. HOMOLOGY THEORY Let F : G E be a homomorphism of an Abelian group G into an Abelian group E . The set of elements of G mapped to the identity element of E is denoted by Ker F = { g G | F ( g ) = 0 E } , where 0 E is the identity element of E , and called the kernel of the homo- morphism F . The image of the homomorphism F : G E is the set Im F = { h E | h = F ( g ) for some g G } . A homomorphism F : G E of a group G into a group E is called an isomorphism if it is a bijection.
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