{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 191

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 G 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 . x/ D ˛ 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 D 0 for j ¤ i , we have
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ²±±±² G s . n Let G be a finite abelian group. For each prime number p define GŒpŁ D f g 2 G W o.g/ is a power of p g : It is straightforward to check that GŒpŁ is a subgroup of G . Since the order of any group element must divide the order of the group, we have GŒpŁ D f g if p does not divide the order of G . Let p be a prime integer. A group (not necessarily finite or abelian) is called a p-group if every element has finite order p k for some k µ . So GŒpŁ is a p –subgroup of G ....
View Full Document

{[ snackBarMessage ]}