Question about normal subgroup chains

Question

If $H<K<G$ where $H\triangleleft G $ is it then true that $H\triangleleft K$?

I am trying to understand subnormal chains and why it is true that every normal chain is a subnormal chain.

Answer 1

As $H \lhd G$, by definition, $\forall g \in G, gHg^{-1} = H$. As $K \subseteq G$, it is therefore also true that $\forall k \in K, kHk^{-1} = H$, as this simply follows as specific cases of the more general statement above.

Thus, yes, $H \lhd K$.

Answer 2

In general, if $K \leq G$ is an arbitrary subgroup and $H \lhd G$, then $H \cap K \lhd K$.