normality of subgroups is not transitive
Let $G$ be a group. A subgroup^{} $K$ of a subgroup $H$ of $G$ is obviously a subgroup of $G$. It seems plausible that a similar^{} situation would also hold for normal subgroups^{}, but in fact it does not: even when $K\mathrm{\u22b4}H$ and $H\mathrm{\u22b4}G$, it is possible that $K\u22ecG$. Here are two examples:

1.
Let $G$ be the subgroup of orientationpreserving isometries (http://planetmath.org/Isometry) of the plane ${\mathbb{R}}^{2}$ ($G$ is just all rotations and translations^{}), let $H$ be the subgroup of $G$ of translations, and let $K$ be the subgroup of $H$ of integer translations ${\tau}_{i,j}(x,y)=(x+i,y+j)$, where $i,j\in \mathbb{Z}$.
Any element $g\in G$ may be represented as $g={r}_{1}\circ {t}_{1}={t}_{2}\circ {r}_{2}$, where ${r}_{1,2}$ are rotations and ${t}_{1,2}$ are translations. So for any translation $t\in H$ we may write
$${g}^{1}\circ t\circ g={r}^{1}\circ {t}^{\prime}\circ r,$$ where ${t}^{\prime}\in H$ is some other translation and $r$ is some rotation. But this is an orientationpreserving isometry of the plane that does not rotate, so it too must be a translation. Thus ${G}^{1}HG=H$, and $H\mathrm{\u22b4}G$.
$H$ is an abelian group^{}, so all its subgroups, $K$ included, are normal.
We claim that $K\u22ecG$. Indeed, if $\rho \in G$ is rotation by ${45}^{\circ}$ about the origin, then ${\rho}^{1}\circ {\tau}_{1,0}\circ \rho $ is not an integer translation.

2.
A related example uses finite subgroups. Let $G={D}_{4}$ be the dihedral group^{} with eight elements (the group of automorphisms^{} of the graph of the square). Then
$${D}_{4}=\u27e8r,f\mid {f}^{2}=1,{r}^{4}=1,fr={r}^{1}f\u27e9$$ is generated by $r$, rotation, and $f$, flipping.
The subgroup
$$H=\u27e8rf,fr\u27e9=\{1,rf,{r}^{2},fr\}\cong {C}_{2}\times {C}_{2}$$ is isomorphic to the Klein 4group – an identity^{} and 3 elements of order 2. $H\mathrm{\u22b4}G$ since $[G:H]=2$. Finally, take
$$K=\u27e8rf\u27e9=\{1,rf\}\mathrm{\u22b4}H.$$ We claim that $K\u22ecG$. And indeed,
$$f\circ rf\circ f=fr\notin K.$$
Title  normality of subgroups is not transitive 

Canonical name  NormalityOfSubgroupsIsNotTransitive 
Date of creation  20130322 12:49:27 
Last modified on  20130322 12:49:27 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  13 
Author  yark (2760) 
Entry type  Example 
Classification  msc 20A05 
Related topic  NormalIsNotTransitive 