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...
View
Full Document
 Fall '08
 EVERAGE
 Algebra, Normal subgroup, N1 N2

Click to edit the document details