# Subgroup isomorphic to whole group need not be normal

From Groupprops

## Statement

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

## Related facts

## Proof

### Example of the infinite dihedral group

`Further information: infinite dihedral group`

Suppose is the infinite dihedral group, given by:

Suppose is the subgroup of generated by and .

Then, the map that sends to and to is an isomorphism from to . Thus, is isomorphic to . However, is not a normal subgroup of , because conjugation by sends to , which is not in .