College Algebra Exam Review 184

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

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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
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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...
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