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Give an example of $ K \subset H \subset G$, such that $K \triangleleft H$ and $H \triangleleft G$ but $ K \triangleleft G$ is not true.

So we're looking for a group $G$ that has a normal subgroup $H$, where $H$ has a normal subgroup $K$, such that $K$ is not a normal subgroup of $G$.

I tried with no success using $(Z_n,+)$ for various n but, but all subgroups are always going to be abelian.

I figured that using $G=S_n$ and $H=A_n$ (symmetry and alternating groups) would get me somewhere, but I couldn't work it out.

Any ideas?

user113867
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2 Answers2

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$\langle s \rangle \unlhd \langle r^2, s \rangle \unlhd D_8$, each being of index 2, but $\langle s \rangle$ is not normal in $D_8$ (the dihedral group of order 8)

zcn
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Consider the cyclic group of order 2 in the Klein 4 group. The Klein 4 is normal in $A_4$, yet the cyclic group is not normal in $A_4$.

Vladhagen
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  • Although I see that this is exactly the example that the above link uses. http://math.stackexchange.com/questions/381035/normal-subgroup-of-a-normal-subgroup – Vladhagen Jan 06 '14 at 22:41