I spend much time to study a theorem - Let G be a nonzero free abelian group of finite rank n, and let K be a nonzero subgroup of G. Then K is free abelian of rank s smaller or equal to n. Furthermore, there exists a basis (x1,x2,....,xn) for G and positive integers d1,d2,...,ds where di divides d(i+1) for i=1,....s-1, such that (d1x1,d2x2,.....dsxs) is a basis for K.(Theorem 4.19 on page 253, Fifth edition, A First Course In Abstract Algebra, by John B. Fraleigh) and I still do not understand the proof at all. Could anyone help me by explaining the proof in more detail ,elaborate and illustrate the theorem by examples.(adsbygoogle = window.adsbygoogle || []).push({});

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# Theorem concerning free abelian groups

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