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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 mathematical 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 notation 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.
 Fall '08
 EVERAGE
 Algebra, Addition

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