Subgroup isomorphic to whole group need not be normal

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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


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.