College Algebra Exam Review 143

College Algebra Exam Review 143 - Q A i Q A i D f e g and Q...

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3.1. DIRECT PRODUCTS 153 normal subgroups of order 2 ; the intersection of any two is f e g and the product of any two is G . G is not isomorphic to the direct product of the three normal subgroups (i.e., G is not isomorphic to Z 2 ± Z 2 ± Z 2 .) Exercises 3.1 3.1.1. Show that A ± B is a group with identity .e A ;e B / and with the inverse of an element .a;b/ equal to .a ± 1 ;b ± 1 / . 3.1.2. Verify that the two subgroups A ±f e B g and f e a B are normal in A ± B , and that the two subgroups have trivial intersection. 3.1.3. Verify that ± 1 W .a;b/ 7! a is a surjective homomorphism of A ± B onto A with kernel f e A g ± B . Likewise, ± 2 W .a;b/ 7! b is a surjective homomorphism of A ± B onto B with kernel A ± f e B g . Conclude that .A ± B/=.A ± f e B g / Š B . 3.1.4. Verify that the coordinate–by–coordinate multiplication on the Carte- sian product P D A 1 ± ²²² ± A n of groups makes P into a group. 3.1.5. Verify that the Q A i D f e g ± ::: f e g ± A i ± f e g ± ::: f e g is a normal subgroup of P D A 1 ± ²²² ± A n , and A i Š Q A i . 3.1.6. Verify that Q A 0 i D A 1 ± ²²² A i ± 1 ± f e g ± A i C 1 ± ²²² ± A n is a normal subgroup of P D A 1 ± ²²² ± A n , that
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Unformatted text preview: Q A i \ Q A i D f e g , and Q A i Q A i D P . 3.1.7. Consider a direct product of groups P D A 1 ± ²²² ± A n . For each i , deﬁne ± i W P ! A i by ± i W .a 1 ;a 2 ;:::;a n / 7! a i . Show that ± i surjective homomorphism of P onto A i with kernel equal to Q A i . Show that ker .± 1 / \ ²²² \ ker .± n / D f e g ; and, for all i , the restriction of ± i to \ j ¤ i ker .± j / maps this subgroup onto A i . 3.1.8. Show that the direct product has an associativity property: A ± .B ± C/ Š .A ± B/ ± C Š A ± B ± C: 3.1.9. Show that the direct product of groups A ± B is abelian, if, and only if both groups A and B are abelian. 3.1.10. Show that none of the following groups is a direct product of groups A ± B , with j A j ; j B j > 1 . (a) S 3 . (b) D 5 (the dihedral group of cardinality 10 )....
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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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