subgroups of finite cyclic group
Let be the order of a finite cyclic group . For every positive divisor (http://planetmath.org/Divisibility) of , there exists one and only one subgroup
of order of . The group has no other subgroups.
Proof. If is a generator of and , then generates the subgroup , the order of which is equal to the order of , i.e. equal to . Any subgroup of is cyclic (see http://planetmath.org/node/4097this entry). If , then must have a generator of order ; thus apparently .