2063. PRODUCTS OF GROUPSThe elementary divisors of a finite abelian group can be obtained fromany direct product decomposition of the group with cyclic factors, as illus-trated in the previous example. IfGŠZa1Zan, then the elementarydivisors are the prime power factors of the integersa1; a2; : : : ; an.The invariant factors can be obtained from the elementary divisors bythe following algorithm, which was illustrated in the example:1.Group together the elementary divisors belonging to each primedividing the order ofG, and arrange the list for each prime inweakly 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 largestinvariant 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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