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
This is the end of the preview.
access the rest of the document.