Unformatted text preview: . Similarly, Z 24 Š Z 3 ² Z 8 . Therefore Z 30 ² Z 24 Š Z 5 ² Z 3 ² Z 2 ² Z 3 ² Z 8 . Regroup these factors as follows: Z 30 ² Z 24 Š . Z 5 ² Z 3 ² Z 8 / ² . Z 3 ² Z 2 / Š Z 120 ² Z 6 . This is the invariant factor decomposition of Z 30 ² Z 24 The invariant factors of Z 30 ² Z 24 are 120;6 . Corollary 3.6.7. (a) Let G be an abelian group of order p n , where p is a prime. Then G is a direct product of cyclic groups, G Š Z p n 1 ² ³³³ ² Z p n k ; where n 1 ± n 2 ± ³³³ ± n k , and P i n i D n , (b) The sequence of exponents in part (a) is unique. That is, if m 1 ± m 2 ± ³³³ ± m ` , P j m j D n , and G Š Z p m 1 ² ³³³ ² Z p m ` ; then k D ` and n i D m i for all i . Proof. This is just the special case of the theorem for a group whose order is a power of a prime. n...
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 Fall '08
 EVERAGE
 Algebra, Cyclic group, Decompositions, invariant factor decomposition, invariant factors, irreducible dividing a1, Z5 Z3 Z2

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