College Algebra Exam Review 188

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online