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Unformatted text preview: C i whose order is a power of p is equal to Gp . Proof. Denote by Ap the sum of those C i whose order is a power of p . Then Ap Gp and G is the internal direct product of the subgroups Ap . Since G is also the internal direct product of the subgroups Gp , it follows that Ap D Gp for all p . n Example 3.6.21. Consider G D Z 30 Z 50 Z 28 . Then G . Z 3 Z 2 Z 5 / . Z 25 Z 2 / . Z 4 Z 7 / . Z 4 Z 2 Z 2 / Z 3 . Z 25 Z 5 / Z 7 : Thus G2 Z 4 Z 2 Z 2 , G3 Z 3 , G5 Z 25 Z 5 , and G7 Z 7 . Gp D for all other primes p . Theorem 3.6.22. (Fundamental Theorem of Finitely Generated Abelian Groups: Elementary Divisor Form). Every nite 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

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