Let be a finite abelian group of order and let be the largest power of the prime dividing . Prove that is isomorphic to the Sylow -subgroup of .
We know that, if is a finite abelian group of order then considered as a -module, is annihilated by , the -primary component of is isomorphic to the direct product of its Sylow subgroups.
Here, is a finite abelian group of order . As a -module, is the direct sum of its Sylow -subgroups, say where is the finite set of primes dividing . Tensor product commute with finite direct sums, so as -homomorphism.
We show if , then . Since and are relatively prime (in this case) in the PID , then there are s.t. . Let be a simple tensor in . Then, . Hence, .
Now we show . Here, every simple tensor in can be written as . Then any tensor, . Now define by . Here, if divides then . Hence is well-defined. It is easy to see that is a -balanced map, so induces a group homomorphism and is the necessary natural inclusion and .
Let then .
It is easy to see that is injective. Also, is surjective, since for each , . Thus is an isomorphism, i.e. .