College Algebra Exam Review 184

College Algebra Exam Review 184 - a 3 C t 4 a 4 3.6...

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

View Full Document Right Arrow Icon
194 3. PRODUCTS OF GROUPS Since the w j ’s are all nonzero, the d j ’s are all nonzero. But then, since f v 1 ;:::;v s g is linearly independent, it follows that f d 1 v 1 ;:::;d s v s g is linearly independent. Thus f w 1 ;:::;w s g D f d 1 v 1 ;:::;d s v s g is a basis of N . n Exercises 3.5 3.5.1. Consider Z as a subgroup of Q . Show that Z is not complemented; that is, there is no subgroup N of Q such that Q D Z ± N . 3.5.2. Show that Q is not a free abelian group. 3.5.3. Show that conditions (a) and (c) in Proposition 3.5.2 are equivalent. 3.5.4. Show that A subset of Z n is linearly independent over Z if, and only if, it is linearly independent over Q . 3.5.5. Show that Z n and Z m are nonisomorphic if m ¤ n . 3.5.6. Modify the proof of Proposition 3.5.7 to show that any subgroup of Z n is free, of rank no more than n . 3.5.7. Let N be a subgroup of Z n , and let f x 1 ;:::;x s g be a generating set for N . Let Q be invertible in Mat s . Z / . Show that the columns of the n –by– s matrix Œx 1 ;:::;x s ŁQ generate N . 3.5.8. Compute the greatest common divisor d of a 1 D 290692787472 , a 2 D 285833616 , a 3 D 282094050438 , and a 4 D 1488 . Find integers t 1 ;t 2 ;t 3 ;t 4 such that d D t 1 a 1 C t 2 a 2 C t 3
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: a 3 C t 4 a 4 . 3.6. Finitely generated abelian groups In this section, we obtain a structure theorem for finitely generated abelian groups. The theorem states that any finitely generated abelian group is a direct product of cyclic groups, with each factor either of in-finite order or of order a power of a prime; furthermore, the number of the cyclic subgroups appearing in the direct product decomposition, and their orders, are unique. Two finite abelian groups are isomorphic if, and only if, they have the same decomposition into a direct product of cyclic groups of infinite or prime power order. The Invariant Factor Decomposition Every finite abelian group G is a quotient of Z n for some n . In fact, if x 1 ;:::;x n is a set of generators of minimum cardinality, we can define a homomorphism of abelian groups from Z n onto G by '. P i r i O e i / D...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online