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College Algebra Exam Review 142

# College Algebra Exam Review 142 - r D e Proof The condition...

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152 3. PRODUCTS OF GROUPS Proposition 3.1.13. Suppose N 1 , N 2 ;:::;N r are normal subgroups of a group G such that for all i , N i \ .N 1 :::N i ± 1 N i C 1 :::N r / D f e g : Then N 1 N 2 :::N r is a subgroup of G and .n 1 ;n 2 ;:::;n r / 7! n 1 n 2 :::n r is an isomorphism of P D N 1 ± N 2 ± ²²² ± N r onto N 1 N 2 :::N r . In particular, if N 1 N 2 :::N r D G , then G Š N 1 ± N 2 ± ²²² ± N r . Proof. Because N i \ N j D f e g for i ¤ j , it follows from Proposition 3.1.5 that xy D yx for x 2 N i and y 2 N j . Using this, it is straight- forward to show that the map .n 1 ;n 2 ;:::;n r / 7! n 1 n 2 :::n r is a homo- morphism of P onto N 1 N 2 :::N r . It remains to check that the map is injective. If .n 1 ;n 2 ;:::;n r / is in the kernel, then n 1 n 2 :::n r D e , so for each i , we have n i D .n ± 1 i ± 1 ²²² n ± 1 2 n ± 1 1 /.n ± 1 r ²²² n ± 1 i C 1 / D n ± 1 1 ²²² n ± 1 i ± 1 n ± 1 i C 1 ²²² n ± 1 r 2 N i \ .N 1 :::N i ± 1 N i C 1 :::N r / D f e g : Thus n i D e for all i . n Corollary 3.1.14. Let N 1 ;N 2 ;:::;N r be normal subgroups of a group G such that N 1 N 2 ²²² N r D G . Then G is the internal direct product of N 1 ;N 2 ;:::;N r if, and only if, whenever x i 2 N i for 1 ³ i ³ r and x 1 x 2 ²²² x r D e , then x 1 D x 2 D ²²² D x
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Unformatted text preview: r D e . Proof. The condition is equivalent to N i \ .N 1 :::N i ± 1 N i C 1 :::N r / D f e g for all i (Exercise 3.1.19 ). n Corollary 3.1.15. Let G be an abelian group, with group operation C . Suppose N 1 ;N 2 ;:::;N r are subgroups with N 1 C N 2 C ²²² C N r D G . Then G is the internal direct product of N 1 ;N 2 ;:::;N r if, and only if, whenever x i 2 N i for 1 ³ i ³ r and P i x i D , then x 1 D x 2 D ²²² D x r D . Remark 3.1.16. Caution: When r > 2 , it does not sufﬁce that N i \ N j D f e g for i ¤ j and N 1 N 2 :::N r D G in order for G to be isomorphic to N 1 ± N 2 ± ²²² ± N r . For example, take G to be Z 2 ± Z 2 . G has three...
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