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I am looking for an example of statement "$K$ is normal in $H$, $H$ is characteristic in $G$ then $K$ is not normal in $G$".

As for statement "$K$ is normal in $H$, $H$ is normal in $G$, $K$ not necessarily normal in $G$" I found out an example of group $A_4$.

$G=A_4,\ H=\{e,(12)(34),(13)(24),(14)(23)\},\ K=\{e,(12)(34)\}$. Then $H$ is normal in $G$ and $K$ is normal in $H$. But $K$ is not normal in $G$.

BUT it's not working for above statement as $H=\{e,(12)(34),(13)(24),(14)(23)\}$ is not characteristic.

What else can help me out ?

Shaun
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Shoaib Ur Rehman
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1 Answers1

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Here is another example. Let $G=A_3 \times S_3$, and put $H=A_3 \times A_3$. Then $|G:H|=2$, whence $H$ is a normal subgroup of $G$ and since $H \in Syl_3(G)$, it follows that $H$ char $G$. Now let $K=\{((1),(1)),((123),(123)),((132),(132))\} \subset H$. Then $K$ is cyclic of order $3$ and $K$ is normal in $H$, for $H$ is abelian. However, $((12),(1))^{-1}((123),(123))((12),(1))=((132),(123)) \notin K$, hence $K$ is not normal in $G$.

Nicky Hekster
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