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 saw in this previous exercise that as a -module, is the internal direct sum of its Sylow subgroups; say , where is the (finite) set of primes dividing . Now tensor products commute with finite direct sums, so that .
We claim that if , then . To see this, let . Since and are relatively prime in the principal ideal domain , there exist integers and such that . Now let be a simple tensor in . We have . Certainly then every element in this tensor product is zero, so that .
Next, we claim that . To see this, note first that every simple tensor (hence every element) in can be written in the form , since for an arbitrary simple tensor we have . Now define by . Note that if divides , then (a-b) \cdot m = 0$, so that . In particular, is well-defined. Evidently, is -balanced, and so induces a group homomorphism . is surjective since . Since and are finite sets, is a bijection and thus a group isomorphism.
Thus we have .