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College Algebra Exam Review 144

College Algebra Exam Review 144 - (b We cannot expect to be...

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154 3. PRODUCTS OF GROUPS (c) D 4 (the dihedral group of cardinality 8 ). (Hint: Use the previous exercise.) 3.1.11. Show that C D R C T . (Recall that C denotes the set of nonzero complex numbers, T the complex numbers of modulus equal to 1. R C denotes the set of strictly positive real numbers.) 3.1.12. Show that GL .n; C / Š C SL .n; C / . (Compare Example 2.7.20 , where it is shown that GL .n; C / D SL .n; C /Z , where Z denotes the group of invertible multiples of the identity matrix.) 3.1.13. Show that Z 8 is not isomorphic to Z 4 Z 2 . (Hint: What is the maximum order of elements of each group?) 3.1.14. Show that Z 4 Z 4 is not isomorphic to Z 4 Z 2 Z 2 . (Hint: Count elements of order 4 in each group.) 3.1.15. Let K 1 be a normal subgroup of a group G 1 , and K 2 a normal subgroup of a group G 2 . Show that K 1 K 2 is a normal subgroup of G 1 G 2 and .G 1 G 2 /=.K 1 K 2 / Š G 1 =K 1 G 2 =K 2 : 3.1.16. Suppose G , A , and B are groups and 1 W G ! A and 2 W G ! B are surjective homomorphisms. Suppose, moreover that ker . 1 / \ ker . 2 / D f e g . (a) Show that W G ! A B , defined by .g/ D . 1 .g/; 2 .g// , is an injective homomorphism of G into A B . Moreover, i ı D i for i D 1; 2 , where i are as in the previous exercise.
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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 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 ex-ercise 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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