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College Algebra Exam Review 196

College Algebra Exam Review 196 - 206 3 PRODUCTS OF GROUPS...

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206 3. PRODUCTS OF GROUPS The elementary divisors of a finite abelian group can be obtained from any direct product decomposition of the group with cyclic factors, as illus- trated in the previous example. If G Š Z a 1 Z a n , then the elementary divisors are the prime power factors of the integers a 1 ; a 2 ; : : : ; a n . The invariant factors can be obtained from the elementary divisors by the following algorithm, which was illustrated in the example: 1. Group together the elementary divisors belonging to each prime dividing the order of G , and arrange the list for each prime in weakly decreasing order. 2. Multiply the largest entries of each list to obtain the largest in- variant factor. 3. Remove the largest entry in each list. Multiply the largest re- maining entries of each non-empty list to obtain the next largest invariant factor. 4. Repeat the previous step until all the lists are exhausted. Example 3.6.24. In the previous example, the lists of elementary divisors,
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