differential geometry w notes from teacher_Part_83

differential geometry w notes from teacher_Part_83 - 165...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

differential geometry w notes from teacher_Part_83 - 165...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online