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 ² A ± A , and deﬁne 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 sufﬁcient condition for to be an isomorphism is that there exist maps ² 1 W A ! G and ² 2 W B ! G satisfying the following conditions: (a) 1 ı ² 1 .a/ D a , while 2 ı ² 1 .a/ D e B for all a 2 A ; and (b) 2 ı ² 2 .b/ D b , while 1 ı ² 2 .b/ D e A for all b 2 B . 3.1.18. (a) Suppose G is a group and ' i W G ³! 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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 Fall '08
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 Algebra, Complex Numbers, Normal subgroup, previous exercise, Z4 Z4

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