Subgroups of a Cyclic Normal Subgroup Are Normal

[e(ho0n3]

Homework Statement
If H ≤ G is cyclic and normal in G, prove that every subgroup of H is also normal in G.

The attempt at a solution
Let H = <h>. We know that for g in G, hⁱ = gh^jg^-1 by the normality of H. A simple induction shows that hⁱⁿ = gh^jng^-1, so that <hⁱ> = g<h^j>g^-1. Now all I need to show is that h^j belongs to <hⁱ>. I'm having trouble with this. Any tips?

 

[mighty2000]

Hello!

Great job on starting the proof. Here are some tips to help you with the rest of the proof:

1. Use the fact that H is cyclic and normal in G: Since H is cyclic, we know that it is generated by a single element, say h. Since H is also normal in G, we know that for any g in G, ghg^-1 is also in H. This means that any element in H can be written as ghg^-1 for some g in G.

2. Use the definition of a normal subgroup: A subgroup N of a group G is normal if for any element g in G, gNg^-1 = N. This means that for any element in N, we can "move" it to the other side of g using the inverse operation. This can help you show that hj belongs to <hi>.

3. Use the fact that <hi> = g<hj>g^-1: This means that for any element in <hi>, we can write it as ghjg^-1 for some g in G. This can also help you show that hj belongs to <hi>.

Hope these tips help you complete the proof! Good luck!