{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 124

College Algebra Exam Review 124 - G=N follows from repeated...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
134 2. BASIC THEORY OF GROUPS The Quotient Group Construction Theorem 2.7.1. Let N be a normal subgroup of a group G . The set of cosets G=N has a unique product that makes G=N a group and that makes the quotient map ± W G ±! G=N a group homomorphism. Proof. Let A and B be elements of G=N (i.e., A and B are left cosets of N in G ). Let a 2 A and b 2 B (so A D aN and B D bN ). We would like to define the product AB to be the left coset containing ab , that is, .aN/.bN/ D abN: But we have to check that this makes sense (i.e., that the result is indepen- dent of the choice of a 2 A and of b 2 B ). So let a 0 be another element of aN and b 0 another element of bN . We need to check that abN D a 0 b 0 N , or, equivalently, that .ab/ ± 1 .a 0 b 0 / 2 N . We have .ab/ ± 1 .a 0 b 0 / D b ± 1 a ± 1 a 0 b 0 D b ± 1 a ± 1 a 0 .bb ± 1 /b 0 D .b ± 1 a ± 1 a 0 b/.b ± 1 b 0 /: Since aN D a 0 N , and bN D b 0 N , we have a ± 1 a 0 2 N and b ± 1 b 0 2 N . Since N is normal, b ± 1 .a ± 1 a 0 /b 2 N . Therefore, the final expression is a product of two elements of N , so is in N . This completes the verification that the definition of the product on G=H makes sense. The associativity of the product on
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: G=N follows from repeated use of the definition of the product, and the associativity of the product on G ; namely .aNbN/cN D abNcN D .ab/cN D a.bc/N D aNbcN D aN.bNcN/: It is clear that N itself serves as the identity for this multiplication and that a ± 1 N is the inverse of aN . Thus G=N with this multiplication is a group. Furthermore, ± is a homomorphism because ±.ab/ D abN D aNbN D ±.a/±.b/: The uniqueness of the product follows simply from the surjectivity of ± : in order for ± to be a homomorphism, it is necessary that aNbN D abN . n The group G=N is called the quotient group of G by N . The map ± W G ! G=N is called the quotient homomorphism. Another approach to defining the product in G=N is developed in Exercise 2.7.2 ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online