Subgroup isomorphic to whole group need not be normal

Statement

It is possible to have a group $G$ and a subgroup $H$ of $G$ such that $H$ and $G$ are isomorphic groups but $H$ is not a Normal subgroup (?) of $G$.

Related facts

• Every group is normal in itself

Proof

Example of the infinite dihedral group

Further information: infinite dihedral group

Suppose $G$ is the infinite dihedral group, given by:

$G:=\langle a,x\mid xax=a^{-1},x=x^{-1}\rangle$

Suppose $H$ is the subgroup of $G$ generated by $a^{4}$ and $x$.

Then, the map $\sigma$ that sends $a$ to $a^{4}$ and $x$ to $x$ is an isomorphism from $G$ to $H$. Thus, $H$ is isomorphic to $G$. However, $H$ is not a normal subgroup of $G$, because conjugation by $a$ sends $x$ to $a^{2}x$, which is not in $H$.