differential geometry w notes from teacher_Part_83

# differential geometry w notes from teacher_Part_83 - 165...

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7.1. ALGEBRAIC PRELIMINARIES 165 Theorem 7.1.1 Fundamental Theorem of Homomorphisms. Let G and F be groups. Let F : G F be a homomorphism. Then G / Ker F ± Im F . Proof : 1. Since F ( g + Ker F ) = F ( g ) . ± Theorem 7.1.2 Let G and E be Abelian groups, and H G and N E be their subgroups so that G / H and E / N are the quotient groups. Let F : G E be a homomorphism such that the image of the subgroup H of G is the subgroup N of E, that is, F ( H ) = N . Then the homomorphism F induces a homomorphism of the quotient groups F * : G / H E / N . Proof : 1. Let π : E E / N be the projection homomorphism deﬁned by, for any x E π ( x ) = [ x ] = x + N . 2. Then F * = π F : G / H E / N is a homomorphism. ± A ﬁeld is a set K with two binary operations, addition, + , and multiplication, · , that satisfy the following conditions: 1. both addition and multiplication are associative, 2. both addition and multiplication are commutative, di geom.tex; April 12, 2006; 17:59; p. 163

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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_83 - 165...

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