Section 9.1 Definition and Examples
Example 9.1.
To show that
Since
the group operation is preserved.
Example 9.2.
We can define an isomorphism
Of course, we must still show that
Example 9.3.
The integers are isomorphic to the subgroup of
By definition the map
Example 9.4.
The groups
An isomorphism
The map
Example 9.5.
Even though
However,
which contradicts the fact that
Theorem 9.6.
Let
is an isomorphism.If
is abelian, then is abelian.If
is cyclic, then is cyclic.If
has a subgroup of order then has a subgroup of order
Proof.
Assertions (1) and (2) follow from the fact that
(3) Suppose that
Theorem 9.7.
All cyclic groups of infinite order are isomorphic to
Proof.
Let
To show that
Theorem 9.8.
If
Proof.
Let
Corollary 9.9.
If
Proof.
The proof is a direct result of Corollary 6.12.
Theorem 9.10.
The isomorphism of groups determines an equivalence relation on the class of all groups.
Subsection Cayley's Theorem
Cayley proved that ifExample 9.11.
Consider the group
The addition table of
Theorem 9.12. Cayley.
Every group is isomorphic to a group of permutations.
Proof.
Let
Hence,
Now we are ready to define our group
We must show that
Also,
and
We can define an isomorphism from
It is also one-to-one, because if
Hence,