College Algebra Exam Review 188

College Algebra Exam Review 188 - 198 3. PRODUCTS OF GROUPS...

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Unformatted text preview: 198 3. PRODUCTS OF GROUPS where A0 is free abelian, and for i 1, Ai Š Zai , where ai 2, and ai divides aj for i Ä j ; and also G D B0 B1 B2 Bt ; where B0 is free abelian, and for i 1, Bi Š Zbi , where bi 2, and bi divides bj for i Ä j ; We have to show that rank.A0 / D rank.B0 /, s D t , and ai D bi for all i 1. By Lemma 3.6.3 , we have Gtor D A1 As D B1 B2 Bt : Hence A0 Š G=Gtor Š B0 . By uniqueness of rank, Proposition 3.5.5, we have rank.A0 / D rank.B0 /. It now suffices to prove that the two decompositions of Gtor are the same, so we may assume that G D Gtor for the rest of the proof. Let a denote the period of G . By Exercise 3.6.6, as D b t D a. We proceed by induction on the length of a, that is, the number of primes (with multiplicity) occuring in an prime factorization of a. If this number is one, then a is prime, and all of the bi and aj are equal to a. In this case, we have only to show that s D t . Since aG D f0g, by Lemma 3.6.5, G is an Za –vector space; moreover, the first direct product decomposition gives G Š Zs and the second gives G Š Zt as Za –vector a a spaces. It follows that s D t by uniqueness of dimension. We assume now that the length of a is greater than one and that the uniqueness assertion holds for all finite abelian groups with a period of smaller length. Let p be prime number. Then x 7! px is a group endomorphism of G that maps each Ai into itself. According to Lemma 3.6.4, if p divides ai then Ai =pAi Š Zp , but if p is relatively prime to ai , then Ai =pAi D f0g. We have G=pG Š .A1 A2 As /=.pA1 pA2 pAs / Š A1 =pA1 A2 =pA2 k As =pAs Š Zp ; where k is the number of ai such that p divides ai . Since p.G=pG/ D f0g, according to Lemma 3.6.5, all the abelian groups in view here are actually Zp –vector spaces and the isomorphisms are Zp –linear. It follows that the number k is the dimension of G=pG as an Zp –vector space. Applying the same considerations to the other direct ...
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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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