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The result you are interested in is Theorem 19 on page 8 of

http://www.math.uga.edu/~pete/4400algebra2point5.pdf

As I explain there, this fact is a kind of duality statement, but it lies deeper than the fact that passage to the dual group takes injections to surjections and conversely (Proposition 16). To deduce Theorem 19 from Proposition 16, one needs the fact that a finite abelian group is [oy vey -- at least] non-canonically isomorphic to its own dual group (Theorem 20), which I go on to prove in Section 5 of these notes in the most elementary way I know how.

Note that the first step in the proof of Theorem 20 develops the Sylow theory of finite abelian groups from scratch -- this is much easier than the nonabelian case.
show/hide this revision's text 2 deleted 21 characters in body

The result you are interested in is Theorem 19 on page 8 of

http://www.math.uga.edu/~pete/4400algebra2point5.pdf

As I explain there, this fact is a kind of duality statement, but it lies deeper than the fact that passage to the dual group takes injections to surjections and conversely (Proposition 16). To deduce Theorem 19 from Proposition 16, one needs the fact that a finite abelian group is non-canonically isomorphic to its own dual group (Theorem 20), which I go on to prove in Section 5 of these notes in the most elementary way I know how.

Note that actually one of the reduction steps first step in the proof of Theorem 20 develops the Sylow theory of finite abelian groups from scratch -- this is much easier than the nonabelian case.
show/hide this revision's text 1

The result you are interested in is Theorem 19 on page 8 of

http://www.math.uga.edu/~pete/4400algebra2point5.pdf

As I explain there, this fact is a kind of duality statement, but it lies deeper than the fact that passage to the dual group takes injections to surjections and conversely (Proposition 16). To deduce Theorem 19 from Proposition 16, one needs the fact that a finite abelian group is non-canonically isomorphic to its own dual group (Theorem 20), which I go on to prove in Section 5 of these notes in the most elementary way I know how.

Note that actually one of the reduction steps in the proof of Theorem 20 develops the Sylow theory of finite abelian groups from scratch -- this is much easier than the nonabelian case.