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College Algebra Exam Review 124

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

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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
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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 ....
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