Is every subgroup of a normal subgroup normal ?
That is if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $H$, then $K$ is a normal subgroup of $G$. Is it true ? If not what is the example?
Progress
$a\in G$ and $k\in K$. Then $k\in H$, since $K\subseteq H$.
Now, $aka^{-1}=k_1aa^{-1}=k_1\in K$ [since $H$ is normal in $G$, $ak=k_1a$]
This implies that $K$ is normal in $H$.
Is my approach correct ?
2 Answers
$\begingroup$The silly counterexample is this: if $H$ is not normal in $G$, then we have $$H \not\lhd G\quad G\lhd G$$ Indeed, this need not even be true if $K$ itself is normal in $H$. For example, in $S_4$, we have $$C_2 \lhd V_4\lhd S_4$$ but $C_2\not\lhd S_4$. (Here, $V_4 = \{(1), (12)(34),(13)(24),(14)(23)\}$ and $C_2 = \{(1), (12)(34)\}$)
The flaw in your argument is taking $ak = k_1 a$ where $k_1\in K$. The fact that $a\in G$ and $H \lhd G$ only allows you to assume that $k_1 \in H$.
$\endgroup$ 0 $\begingroup$G is a normal subgroup of itself, but it might have subgroups that are not normal.
$\endgroup$ 4
