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.
 Fall '08
 EVERAGE
 Algebra

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