College Algebra Exam Review 191

College Algebra Exam Review 191 - D for j i , we have D r i...

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3.6. FINITELY GENERATED ABELIAN GROUPS 201 to Cauchy’s theorem for abelian groups, no prime other than p divides the order of G . n The Primary Decomposition Proposition 3.6.13. Let G be a finite abelian group of cardinality n . Write n D ˛ 1 ˛ 2 ±±± ˛ s , where the ˛ i are pairwise relatively prime natural num- bers, each at least 2. Let G i D f x 2 G W ˛ i x D 0 g . Then G i is a subgroup and G D G 1 ² G 2 ² ±±± ² G s : Proof. If x and y are elements of G i , then ˛ i .x C y/ D ˛ i x C ˛ i y D 0 , and ˛ i . ³ x/ D ³ ˛ i x D 0 , so G i is closed under the group operation and inverses. For each index i let r i D n=˛ i ; that is, r i is the largest divisor of n that is relatively prime to ˛ i . For all x 2 G , we have r i x 2 G i , because ˛ i .r i x/ D nx D 0 . Furthermore, if x 2 G j for some j ¤ i , then r i x D 0 , because ˛ j divides r i . The greatest common divisor of f r 1 ;:::;r s g is 1. Therefore, there exist integers t 1 ;:::;t s such that t 1 r 1 C ±±± C t s r s D 1 . Hence for any x 2 G , x D 1x D t 1 r 1 x C ±±± C t s r s x 2 G 1 C G 2 C ±±± C G s . Thus G D G 1 C ±±± C G s . Suppose that x j 2 G j for 1 ´ j ´ s and P j x j D 0 . Fix an index i . Since r i x j
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Unformatted text preview: D for j i , we have D r i . X j x j / D X j r i x j D r i x i : Because r i is relatively prime to the order of each nonzero element of G i , it follows that x i D . Thus by Proposition 3.5.1 , G D G 1 G s . n Let G be a nite abelian group. For each prime number p dene Gp D f g 2 G W o.g/ is a power of p g : It is straightforward to check that Gp is a subgroup of G . Since the order of any group element must divide the order of the group, we have Gp D f g if p does not divide the order of G . Let p be a prime integer. A group (not necessarily nite or abelian) is called a p-group if every element has nite order p k for some k . So Gp is a p subgroup of G ....
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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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