If $G = \langle x \rangle$ and $H$ is a subgroup of $G$ is $H\unlhd G$?

Question

If $G = \langle x \rangle$ and $H \leq G$, is $H\unlhd G$?

So I know that the group generated by $x$ is known as the generator, and it makes $G$ cyclic iff $G = \langle x \rangle$. If $H \leq G$, can we assume that $H\unlhd G$?

Very lost here on how to prove if a subgroup is normal.

Yes! $G$ is abelian, so every subgroup is normal. – Albert Apr 26 '18 at 03:49 — Albert, Apr 26 '18 at 03:49

score 1 · Answer 1 · answered Apr 26 '18 at 04:18

A subgroup $H\leq G$ is normal if $aH=Ha$ for all $a\in G$.

One way to prove a subgroup is normal is to use the normal subgroup test:

If $H\leq G$ and $aHa^{-1}\subseteq H$ for all $a\in G$ then $H\trianglelefteq G$.

It is not difficult to show any subgroup of an Abelian group is normal. (Proof). Your question is a special case of that fact since any cyclic group is Abelian.

If $G = \langle x \rangle$ and $H$ is a subgroup of $G$ is $H\unlhd G$?

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