Prove that a finite abelian group is (isomorphic to) the direct product of its Sylow subgroups.
For a finite group , we call the number of distinct primes dividing the breadth of . We proceed by induction on the breadth of .
First, note that a Sylow -subgroup of an abelian group is normal, hence unique.
For the base case, if is a finite abelian group of breadth 1, we have . Clearly then is its own (unique) Sylow subgroup, and is trivially isomorphic to the direct product of itself.
For the inductive step, suppose that every finite abelian group of breadth is isomorphic to the direct product of its Sylow subgroups. Let be a finite abelian group of breadth , and say . Denote by the (unique) Sylow -subgroup of . Note that by Lagrange. By the induction hypothesis, . Thus , and by the recognition theorem, .
The conclusion holds for finite abelian groups of arbitrary breadth by induction.