Unformatted text preview: C i whose order is a power of p is equal to GŒpŁ . Proof. Denote by AŒpŁ the sum of those C i whose order is a power of p . Then AŒpŁ ³ GŒpŁ and G is the internal direct product of the subgroups AŒpŁ . Since G is also the internal direct product of the subgroups GŒpŁ , it follows that AŒpŁ D GŒpŁ for all p . n Example 3.6.21. Consider G D Z 30 ´ Z 50 ´ Z 28 . Then G Š . Z 3 ´ Z 2 ´ Z 5 / ´ . Z 25 ´ Z 2 / ´ . Z 4 ´ Z 7 / Š . Z 4 ´ Z 2 ´ Z 2 / ´ Z 3 ´ . Z 25 ´ Z 5 / ´ Z 7 : Thus GŒ2Ł Š Z 4 ´ Z 2 ´ Z 2 , GŒ3Ł Š Z 3 , GŒ5Ł Š Z 25 ´ Z 5 , and GŒ7Ł Š Z 7 . GŒpŁ D for all other primes p . Theorem 3.6.22. (Fundamental Theorem of Finitely Generated Abelian Groups: Elementary Divisor Form). Every ﬁnite abelian group is isomorphic to a direct product of cyclic groups of prime power order. The number...
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 Fall '08
 EVERAGE
 Algebra, Congruence, Abelian group, Cyclic group, finite abelian group, Finitely Generated Abelian

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