Properties of Isomorphism
Theorem 1:
If isomorphism exists between two groups, then the identities correspond, i.e. if is an isomorphism and
are respectively the identities in
, then
.
Theorem 2:
If isomorphism exists between two groups, then the identities correspond, i.e. if is an isomorphism and
, where
then
.
Theorem 3:
In an isomorphism the order of an element is preserved, i.e. if is an isomorphism, and the order of
is
, then the order of
is also
.
Proof:
As , then we have
and in general we can write it as
.
But , by using the statement of Theorem 1,
therefore . Also
for
, i.e.
.
It follows that the order of an element of , if finite, is equal to the order of its image in
. If the order of
is infinite, we can similarly show that the order of
cannot be finite.
Theorem 4:
The relation of isomorphism in the set of groups is an equivalence relation.