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College Algebra Exam Review 176

College Algebra Exam Review 176 - 186 3 PRODUCTS OF GROUPS...

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186 3. PRODUCTS OF GROUPS ' W .a 1 ;:::;a n / 7! P i a i x i . The kernel N of ' is a subgroup of Z n , and G Š Z n =N . Conversely, for any subgroup N of Z n , Z n =N is a finitely generated abelian group. Therefore, to understand finitely generated abelian groups, we must understand subgroups of Z n . The study of subgroups of Z n involves linear algebra over Z . The theory presented in this section and the next is a special case of the structure theory for finitely generated modules over a principal ideal domain, which is discussed in sections 8.4 and 8.5 , beginning on page 369 . The reader or instructor who needs to save time may therefore prefer to omit some of the proofs in Sections 3.5 and 3.6 in the expectation of treating the general case in detail. We define linear independence in abelian groups as for vector spaces: a subset S of an abelian group G is linearly independent over Z if whenever x 1 ;:::;x n are distinct elements of S and r 1 ;:::;r n are elements of Z , if r 1
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