# Thread: Cyclic subgroups

1. ## Cyclic subgroups

If $X$ is a cyclic normal subgroup of $Y$ I need to prove that any subgroup of $X$ is also a normal subgroup of $Y$.

This seems trivial but I'm not sure that it is.

2. Originally Posted by d_p_osters
If $X$ is a cyclic normal subgroup of $Y$ I need to prove that any subgroup of $X$ is also a normal subgroup of $Y$.

This seems trivial but I'm not sure that it is.
If X has generator a, and Z is a subgroup of X, generated by $a^k$, then Z is the set of all elements of the form $a^{nk}$.

If $g\in G$, then $g^{-1}Zg$ is generated by $g^{-1}a^kg = (g^{-1}ag)^k$, from which you can see that $g^{-1}ZG\subseteq Z$.