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Unformatted text preview: . Similarly, Z 24 Z 3 Z 8 . Therefore Z 30 Z 24 Z 5 Z 3 Z 2 Z 3 Z 8 . Regroup these factors as follows: Z 30 Z 24 . Z 5 Z 3 Z 8 / . Z 3 Z 2 / Z 120 Z 6 . This is the invariant factor decomposition of Z 30 Z 24 The invariant factors of Z 30 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 Z p n 1 Z p n k ; where n 1 n 2 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 m ` , P j m j D n , and G Z p m 1 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

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