# 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 $G$.

## 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 $a^2x$, which is not in $H$.