Let be a finite abelian group of order and consider as a -module in the natural way. Prove that is annihilated by , that the -primary component of is the (unique) Sylow -subgroup of , and that is isomorphic to the direct product of its Sylow subgroups.
Certainly for all , . Thus . Since is a principal ideal domain, by this previous exercise, , where .
We claim that each is a Sylow subgroup of . To that end, note that ; in particular, every element of has -power order. Conversely, if has -power order, then . Hence the , that is, the -primary components, are Sylow subgroups of . Since is abelian, there is a unique Sylow -subgroup for each ; thus is the internal direct sum of its Sylow subgroups. Since the direct sum is finite, is in fact the internal direct product of its Sylow subgroups.