College Algebra Exam Review 195

College Algebra Exam Review 195 - M D N and fj C i j W 1 i...

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3.6. FINITELY GENERATED ABELIAN GROUPS 205 of cyclic groups of each order appearing in such a direct product decom- position is uniquely determined. Proof. We have already have observed that a finite abelian group G is isomorphic to a direct product of cyclic groups of prime power order. We need to verify the uniquenes of the orders of the cyclic groups appearing in such a direct product decompostion. Suppose f C i W 1 ± i ± N g and f D j W 1 ± j ± M g are two families of cyclic subgroups of G of prime power order such that G D C 1 ² ³³³ ² C N D D 1 ² ³³³ ² D M : Group each family of cyclic subgroups according to the primes dividing j G j , f C i g D [ p f C p i W 1 ± i ± N.p/ g ; and f D j g D [ p f D p j W 1 ± j ± M.p/ g ; where each C p i and D p j has order a power of p . According to the previous lemma, P N.p/ i D 1 C p i D P M.p/ j D 1 D p j D GŒpŁ for each prime p dividing j G j . It follows from Corollary 3.6.7 and Corollary 3.6.15 that N.p/ D M.p/ and fj C p i j W 1 ± i ± N.p/ g D fj D p j j W 1 ± j ± N.p/ g : It follows that
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Unformatted text preview: M D N and fj C i j W 1 i N g D fj D j j W 1 j N g : n The direct product decomposition of a nite abelian group with factors cyclic groups of prime power order is called the elementary divisor decom-position. The orders of the factors are called the elementary divisors of G . Example 3.6.23. Consider the example G D Z 30 Z 50 Z 28 again. The the elementary divisor decomposition of G is: G . Z 4 Z 2 Z 2 / Z 3 . Z 25 Z 5 / Z 7 : The elementary divisors are 4;2;2;3;25;5;7 . The invariant factor dec-composition can be obtained regrouping the factors as follows: G . Z 4 Z 3 Z 25 Z 7 / . Z 2 Z 5 / Z 2 Z 4 3 25 7 Z 2 5 Z 2 Z 2100 Z 10 Z 2 :...
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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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