Cyclic Normal Groups: Proving Normality of Subgroups in Cyclic Groups"

[math8]

Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G.

I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have
K char in H and H normal in G.
Hence K normal in G.

I am not sure about the part where subgroups of cyclic groups are characteristic. If yes, How would you prove this?

 

[Dick]

Think concretely. A cyclic group is isomorphic to either the integers Z, or the integers mod n, Z_n. Can you prove any subgroup of those is characteristic?

 

[math8]

I am thinking maybe that since cyclic groups of a certain order are unique up to isomorphism and that if a subgroup K of a certain order is unique in a group H, then K char in G.

Now since K is cyclic in H, then K char in H.

 

[Dick]

I am thinking maybe that since cyclic groups of a certain order are unique up to isomorphism and that if a subgroup K of a certain order is unique in a group H, then K char in G.

Now since K is cyclic in H, then K char in H.

Something like that. If you can prove there is exactly one subgroup of a given order in Z_n then you've got it. In the infinite case of Z, it's not going to be useful to consider order though.