Ifis an isomorphism from a group
onto a group
then
1.so
preserves the identity.
This is a consequence of the morphism propertyTake
then
Cancelling
from both sides gives
2.This is proved by induction on the morphism property. Note that
Suppose that
then
This relation implies that order is preserved, since ifand this solution cannot be satisfied for
since
would not then be one to one. It also implies that
is cyclic if and if only if
is cyclic.
3. Forand
with
and
if and only if
Supposethen
and vice versa.
A consequence is thatis abelian if and only if
is 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 implies
by the morphism property.