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Unformatted text preview: (b) We cannot expect to be surjective in general, even though 1 and 2 are surjective. For example, take A D B , take G to be the diagonal subgroup , G D f .a;a/ W a 2 A g A A , and dene i W G ! A by i ..a;a// D a for i D 1;2 . Show that W G ! A B is just the inclusion of G into A B , which is not surjective. 3.1.17. What sort of conditions on the maps 1 , 2 in the previous exercise will ensure that W G ! A B is an isomorphism? Show that a necessary and sufcient condition for to be an isomorphism is that there exist maps 1 W A ! G and 2 W B ! G satisfying the following conditions: (a) 1 1 .a/ D a , while 2 1 .a/ D e B for all a 2 A ; and (b) 2 2 .b/ D b , while 1 2 .b/ D e A for all b 2 B . 3.1.18. (a) Suppose G is a group and ' i W G ! A i is a homomorphism for i D 1;2;:::;r . Suppose ker .' 1 / \ ::: ker .' r / D f e g . Show...
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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, Complex Numbers

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