Let G be a finite cyclic group prove that every subgroup H of G is isomorphic to some factor group G mod K and every factor group G mod K is isomorphic to some subgroup H?

Let ( G ) be a finite cyclic group generated by an element ( g ). Any subgroup ( H ) of ( G ) can be expressed as ( H = \langle g^k \rangle ) for some divisor ( k ) of the order of ( G ). The factor group ( G/K ) for some subgroup ( K ) is also cyclic, and by choosing ( K ) appropriately (for instance, ( K = \langle g^m \rangle ) where ( m ) divides the order of ( G )), we can ensure that ( G/K ) is isomorphic to ( H ). Thus, every subgroup ( H ) corresponds to a factor group ( G/K ) and vice versa, establishing the desired isomorphism relationships.