| Version 19 |
Version 18 |
| \PMlinkescapeword{additive} |
\PMlinkescapeword{additive} |
| \PMlinkescapeword{homomorphism} |
\PMlinkescapeword{homomorphism} |
| \PMlinkescapeword{module} |
\PMlinkescapeword{module} |
| \PMlinkescapeword{modules} |
\PMlinkescapeword{modules} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
| \PMlinkescapeword{subgroup} |
\PMlinkescapeword{subgroup} |
| \PMlinkescapeword{subgroups} |
\PMlinkescapeword{subgroups} |
| \PMlinkescapeword{theorem} |
\PMlinkescapeword{theorem} |
| \PMlinkescapeword{unitary} |
\PMlinkescapeword{unitary} |
| \PMlinkescapeword{word} |
\PMlinkescapeword{word} |
|
|
| Let $(G,*)$ be a group. If for any $a,b\in G$ we have |
Let $(G,*)$ be a group. If for any $a,b\in G$ we have |
| $a*b=b*a$, we say that the group is \emph{abelian} (or \emph{commutative}). |
$a*b=b*a$, we say that the group is \emph{abelian} (or \emph{commutative}). |
| Abelian groups are named after Niels Henrik Abel, but the word {\it abelian} is commonly written in lowercase. |
Abelian groups are named after Niels Henrik Abel, but the word {\it abelian} is commonly written in lowercase. |
|
|
| Abelian groups are essentially the same thing as unitary $\Z$-\PMlinkname{modules}{Module}. |
Abelian groups are essentially the same thing as unitary $\Z$-\PMlinkname{modules}{Module}. |
| In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation. |
In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation. |
|
|
| Some of the basic properties of abelian groups are as follows: |
Some of the basic properties of abelian groups are as follows: |
|
|
| \begin{thm} |
\begin{thm} |
| Any \PMlinkname{subgroup}{Subgroup} of an abelian group is normal. |
Any \PMlinkname{subgroup}{Subgroup} of an abelian group is normal. |
| \end{thm} |
\end{thm} |
| \begin{proof} |
\begin{proof} |
| Let $H$ be a subgroup of the abelian group $G$. Since $ah=ha$ for any $a\in G$ and any $h\in H$ we get $aH=Ha$. That is, $H$ is normal in $G$. |
Let $H$ be a subgroup of the abelian group $G$. Since $ah=ha$ for any $a\in G$ and any $h\in H$ we get $aH=Ha$. That is, $H$ is normal in $G$. |
| \end{proof} |
\end{proof} |
|
|
| \begin{thm} |
\begin{thm} |
| Quotient groups of abelian groups are also abelian. |
Quotient groups of abelian groups are also abelian. |
| \end{thm} |
\end{thm} |
| \begin{proof} |
\begin{proof} |
| Let $H$ a subgroup of $G$. Since $G$ is abelian, $H$ is normal and we can get the quotient group $G/H$ whose elements are the equivalence classes for |
Let $H$ a subgroup of $G$. Since $G$ is abelian, $H$ is normal and we can get the quotient group $G/H$ whose elements are the equivalence classes for |
| $a\sim b$ if $ab^{-1}\in H$. |
$a\sim b$ if $ab^{-1}\in H$. |
| The operation on the quotient group is given by $aH\cdot bH=(ab)H$. But $bH\cdot aH=(ba)H =(ab)H$, therefore the quotient group is also commutative. |
The operation on the quotient group is given by $aH\cdot bH=(ab)H$. But $bH\cdot aH=(ba)H =(ab)H$, therefore the quotient group is also commutative. |
| \end{proof} |
\end{proof} |
|
|
| Here is another theorem concerning abelian groups: |
Here is another theorem concerning abelian groups: |
|
|
| \begin{thm} |
\begin{thm} |
| If $\varphi\colon G\to G$ defined by $ \varphi(x) =x^2$ is a \PMlinkname{homomorphism}{GroupHomomorphism}, then $G$ is abelian. |
If $\varphi\colon G\to G$ defined by $ \varphi(x) =x^2$ is a \PMlinkname{homomorphism}{GroupHomomorphism}, then $G$ is abelian. |
| \end{thm} |
\end{thm} |
|
\begin{proof}
|
\begin{roof}
|
| If such a function were a homomorphism, |
If such a function were a homomorphism, |
| we would have |
we would have |
| \[(xy)^2=\varphi(xy) = \varphi(x)\varphi(y)=x^2y^2\] that is, $xyxy=xxyy$. |
\[(xy)^2=\varphi(xy) = \varphi(x)\varphi(y)=x^2y^2\] that is, $xyxy=xxyy$. |
| Left-multiplying by $x^{-1}$ and right-multiplying by $y^{-1}$ we are led to |
Left-multiplying by $x^{-1}$ and right-multiplying by $y^{-1}$ we are led to |
| $yx=xy$ and thus the group is abelian. |
$yx=xy$ and thus the group is abelian. |
| \end{proof} |
\end{proof} |
