Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms.
The Isomorphism Theorems
Although it is not evident at first,
factor groups correspond exactly to homomorphic images,
and we can use factor groups to study homomorphisms.
We already know that with every group homomorphism
\(\phi: G \rightarrow H\) we can associate a normal subgroup of \(G\),
\(\ker \phi\).
The converse is also true; that is,
every normal subgroup of a group \(G\) gives rise to homomorphism of groups.
Let \(H\) be a normal subgroup of \(G\).
Define the natural or
canonical homomorphism
\[\begin{aligned}\end{aligned}\]
by
\[\begin{aligned}\end{aligned}\].
This is indeed a homomorphism, since
\[\begin{aligned}\end{aligned}\].
The kernel of this homomorphism is \(H\).
The following theorems describe the relationships between group homomorphisms,
normal subgroups, and factor groups.
Mathematicians often use diagrams called
commutative diagrams
to describe such theorems.
The following diagram commutes
since \(\psi = \eta \phi\).
Notice that in the course of the proof of , we have also proved the following theorem.
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.