Let $N$ be a subgroup of $G$, One can show that,

$N$ is normal in $G$ if and only if for all $xy\in N$, $yx\in N$.

Above proposion can be proved by elemantary methods.

I wonder the following;

Let $H$ be a subgroup of $G$ with following property,

If $xyz \in H$ then all possible product of $x,y,z$ in $H$, i.e $xzy,yxz,zyx..\in H$.

By considering above argument, we can directly say that $H$ is normal in $G$ but actually it is stronger than normality so I wonder whether such $H$ must be characteristic or not or what can we say about $H$ in that case ?