College Algebra Exam Review 176

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

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

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the 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