College Algebra Exam Review 103

College Algebra - homomorphism Proof Exercise 2.4.3 n Next we check that homomorphisms preserve the group identity and inverses Proposition 2.4.11

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2.4. HOMOMORPHISMS AND ISOMORPHISMS 113 Example 2.4.8. There is a homomorphism from Z to Z n deﬁned by k 7! ŒkŁ . This follows directly from the deﬁnition of the addition in Z n : ŒaŁ C ŒbŁ D Œa C . (But recall that we had to check that this deﬁnition makes sense; see the discussion following Lemma 1.7.5 .) This example is also a special case of the previous example, with G D Z n and the chosen element a D Œ1Ł 2 G . The map is given by k 7! kŒ1Ł D ŒkŁ . Example 2.4.9. Let G be an abelian group and n a ﬁxed integer. Then the map from G to G given by g 7! g n is a group homomorphism. This is equivalent to the statement that .ab/ n D a n b n when a;b are elements in an abelian group. Let us now turn from the examples to some general observations. Our ﬁrst observation is that the composition of homomorphisms is a homomor- phism. Proposition 2.4.10. Let ' W G ±! H and W H ±! K be homo- morphisms of groups. Then the composition ı ' W G ±! K is also a
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Unformatted text preview: homomorphism. Proof. Exercise 2.4.3 . n Next we check that homomorphisms preserve the group identity and inverses. Proposition 2.4.11. Let ' W G ±! H be a homomorphism of groups. (a) '.e G / D e H . (b) For each g 2 G , '.g ± 1 / D .'.g// ± 1 . Proof. For any g 2 G , '.e G /'.g/ D '.e G g/ D '.g/: It follows from Proposition 2.1.1 (a) that '.e G / D e H . Similarly, for any g 2 G , '.g ± 1 /'.g/ D '.g ± 1 g/ D '.e G / D e H ; so Proposition 2.1.1 (b) implies that '.g ± 1 / D .'.g// ± 1 . n Before stating the next proposition, we recall some conventional math-ematical notation. For any function f W X ±! Y , and any subset B ² Y , the preimage of B in X is f x 2 X W f.x/ 2 B g . The conventional no-tation for the preimage of B is f ± 1 .B/ . The preimage of B makes sense...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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