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## MODERN ALGEBRA

by: Estevan Champlin Sr.

118

0

1

# MODERN ALGEBRA MATH 220A

Estevan Champlin Sr.
UCSB
GPA 3.66

Staff

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COURSE
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1
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KARMA
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## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Estevan Champlin Sr. on Thursday October 22, 2015. The Class Notes belongs to MATH 220A at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 118 views. For similar materials see /class/226883/math-220a-university-of-california-santa-barbara in Mathematics (M) at University of California Santa Barbara.

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Date Created: 10/22/15
MATH 220A NOVEMBER 26 2008 The Structure Theorem for Finitely Presented Abelian Groups In the previous handout we explained the mechanics of the Smith Normal Form algorithm In this handout we discuss in more detail how that algorithm is applied to nd the structure of nitely presented abelian groups If S is a set the free abelian group on S denoted by Z5 is the set of all linear combinations 2565 ass with aS S Z where we restrict to combinations for which aS 0 for all but nitely many 5 S S The group law is Z ass Z bss 2m b9s SSS SSS SSS A group G is nitely generated if there is an epimorphism 7T Z5 a G for some nite set S If the kernel of 7139 is also nitely generated we say that G is nitely presented It is not hard to see that any nite group is nitely presented For a nitely presented group there is a homomorphism p ZT a Z5 for some nite sets S and T such that the image of pis the kernel of 7139 Thus the original group G is isomorphic to the cohernel of p G 2 ZS im p There may be many di erent generating sets for G and for ker 7139 We will restrict our attention to changes of generating sets for Z5 and ZT where we take the sets S and T to be nite lf Z5 2 Z5 for two sets S 51 sm and S sisn of the same cardinality then there must be expressions 7 E 7 2 8739 7 810 8 i SkUM which can be summarized by two square matrices 2 and 2 with integer entries by de nition we have i i s 7 50 7 skakiaw so 22 Im the m gtlt in identity matrix Similarly 22 Im In other words changing the generating set for Z5 is accomplished via an invertible m gtlt m matrics over the integers Similarly changing the generating set for ZT is accomplished Via an invertible n gtlt n matrix over the integers If we have chosen generating sets S 51 sm and T t1 tn then the homomorphism p can be described by means of its action on the generators we can write PW 2 Sim 1

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