This is Cauchy's theorem restricted to abelian groups.
Assume then that the statement has been show for groups of order and let .
If does have a non-trivial proper subgroup then divides either or .
In the first case by induction assumption has an element of order which is therefore also an element of of order .
In the second case there is by induction assumption an element such that has order . Since the order of divides the order of it follows that has order for some . Then has order .
See for instance (Sullivan).
A new proof of the fundamental theorem of finite abelian groups was given in