Subgroup isomorphic to whole group need not be normal
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 .