A subgroup of a cyclic normal subgroup of a Group is Normal

Question

Is it true that Subgroup of a Cyclic Normal subgroup of a Group is again Normal ? If so any hints for the proof?

Answer 1

Sure. It follows from a more general fact: a characteristic subgroup of a normal subgroup of $G$ is also a normal subgroup of $G$. It's even easier to think about the question in these general terms.

Answer 2

If $H<N\lhd G$ and $N$ is cyclic, then $H$ is a characteristic subgroup of $N$, i.e. it is left invariant by all automorphism of $N$, thus including those outer automorphisms of $N$ that are actually inner automorhisms of $G$, i.e. $H$ is conjugation invariant.

Answer 3

Yes. This is one of the several cases when normality is transitive. If

$$N\triangleleft C\triangleleft G\,\,\,,\,\,C=\text{cyclic, then}\;\; C=\langle c\rangle\Longrightarrow N=\langle c^k\rangle\Longrightarrow$$

$$\Longrightarrow \,\,\forall g\in G\,\,,\,\,x^{-1}c^{rk}x=(x^{-1}c^kx)^r\in N\,....$$