Definition.
Let $G$ be a group with operation $*$
and let $H$ be a group with operation
$\hash.$ An
isomorphism of $G$ onto $H$
is a mapping $\theta:G\rightarrow H$
that is one-to-one and onto and satisfies,
for all $a,b\in G,$
\[
\theta(a*b)=\theta(a)\hash\theta(b).
\]
Definition.
If there is an isomorphism of $G$ onto $H,$
then $G$ and $H$ are said to be
isomorphic,
and we write $G\approx H.$
The condition
\(
\theta(a*b)=\theta(a)\hash\theta(b)
\)
is sometimes described by saying that
$\theta$
preserves the operation.
Note.
It makes no difference whether we
operate in $G$ and then apply $\theta$
or apply $\theta$ first and then
operate in $H,$ we get the same
result either way.
Example 18.1.
The obvious mapping from Arabic to
Roman numerals is an isomorphism.
Example 18.2.
The following mapping from
$\gen{(1\ 2 \ 3)}$
to $\Z_3,$ is an isomorphism.
The elements and
operation of the former are
permutations and composition,
the latter, congruence classes and
addition modulo 3.
\begin{align*}
(1)&\mapsto[0]\\
(1\ 2\ 3)&\mapsto[1]\\
(1\ 3\ 2)&\mapsto[2]
\end{align*}
Example 18.3.
The mapping from the set of all
integers with addition, to the set
of even integers with addition,
is an isomorphism.
Example 18.4.
The function $\log=\log_{10},$
from the positive reals with
multiplication, to the reals
with addition, is an isomorphism,
since
\[
\log(xy)=\log{x}+\log{y}.
\]
Theorem 18.1.
If $G$ and $H$ are isomorphic groups
and $G$ is Abelian, then $H$ is Abelian.
Definition.
If $G$ and $H$ are groups with operations
$*$ and $\hash,$ respectively,
then $\theta:G\rightarrow H$ is
a
homomorphism
if, for all $a,b\in G,$
\[
\theta(a*b)=\theta(a)\hash\theta(b)
\]
Guevara Corollary.
Thus, an isomorphism is a homomorphism that is
one-to-one and onto.
Theorem 18.2.
Let $G$ and $H$ be groups with operations
$*$ and $\hash,$ respectively,
and $\theta:G\rightarrow H$ be a
homomorphism. Then
-
\(
\theta(e_G)=e_H
\)
-
\(
\theta(a^{-1})=\theta(a)^{-1}
\)
for each $a\in G.$
-
\(
\theta(a^k)=\theta(a)^k
\)
for each $a\in G$ and integer $k.$
-
\(
\theta(G)
\)
the image of $\theta,$
is a subgroup of $H,$ and
-
if $\theta$ is one-to-one, then
\(
G\approx\theta(G).
\)
Alex Notes