Is the normality of a subgroup dependent on which group is its parent?

Question

It is important to understand the relationship of normal subgroups to their parent. One concept that needs to be understood is whether the normality of a subgroup does not depend on which parent group it is a subgroup of. This is true for some subgroups ($\{e\}$ is always normal in any group) but is it generally speaking true for every group? The question can be phrased in symbolic form: let $G, K$ be groups and $H \leq G$ and $H \leq K$. If $H \lhd G$ then necessarily $H \lhd K$?

Answer 1

Yes.

$S_4$ has a Sylow-3 subgroup $\mathbb{Z}_3$, but it is not normal because the number of Sylow-3 subgroups $> 1$. And a Sylow-p subgroup is normal iff. $n_p = 1$. $S_3$ has $\mathbb{Z}_3$ as a normal Sylow-p subgroup for the same reason.

Answer 2

Normality is relative to the parent group. This is evident in the way it is defined. For a group $G$ and subgroup $N$ of $G$ we say that $N$ is normal in $G$ if for all $g \in G$ we have $N^g=N$. What brings the parent group in the definition is the "if for all $g \in G$" bit.

To give a concrete example: every group is a normal subgroup of itself, clearly. Now let $G$ be a non-abelian finite simple group and let $H$ be a proper and non-trivial subgroup of $G$. Then $H$ is normal in itself, but it is not normal in $G$.