# Counter Example for Normality of Groups

Give a Counter Example to show that if $K \unlhd H$ and $H$ is a characteristic of $G$, then K need not be normal in $G$.

I have no idea which type of groups I should look at? Any hint is appreciated!

• Hint: Think about $G = S_{4}.$ – Geoff Robinson Mar 16 '15 at 20:05
• – vadim123 Mar 16 '15 at 20:06
• You know that it will need to be a non-abelian group. As a general strategy, whenever you have this requirement, look at a small symmetric group. $S_4$ is a good one, as @GeoffRobinson mentioned. – rnrstopstraffic Mar 16 '15 at 20:07