If N is cyclic normal in G and H is a subgroup of N, is H normal in G?

[moont14263]

Please can someone help me to prove this
If N is a cyclic normal subgroup of G and H is a subgroup of N then H is normal in G.

 

[micromass]

You'll need to use the following:

If x in N has order k, then gxg^-1 also has order n.

So conjugation won't change the order of the elements, and order of the elements determine which subgroup the element will belong to (in cyclic groups).

 

[moont14263]

Could you give me more hints because I am not that much in cyclic groups. Thank you.

 

[micromass]

Show that, if G is abelian and if n is a number, then $$G(n)=\{g\in G~\vert~g^n=e\}$$ is a subgroup of G.

Then show that, if G is cyclic, then all subgroups of G are of the form G(n).