Properties of Isomorphisms
Ifis an isomorphism from a grouponto a groupthen
1.sopreserves the identity.
This is a consequence of the morphism propertyTakethenCancellingfrom both sides gives
2.This is proved by induction on the morphism property. Note thatSuppose thatthen
This relation implies that order is preserved, since ifand this solution cannot be satisfied forsincewould not then be one to one. It also implies thatis cyclic if and if only ifis cyclic.
3. Forandwithandif and only if
Supposethen and vice versa.
A consequence is thatis abelian if and only ifis abelian.
4. The number of solutions is preserved – a consequence of the morphism property, and being one to one.
5. The inverseis an isomorphism from H onto G.
6. The property of being a subgroup or a normal subgroup is preserved.
7. The property of elements being conjugate is preserved since impliesby the morphism property.