| Version 8 |
Version 7 |
| \PMlinkescapeword{properties} |
Let $(G,*)$ be a group. If for any $a,b\inG$ we have |
| \PMlinkescapeword{subgroup} |
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| \PMlinkescapeword{subgroups} |
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| Let $(G,*)$ be a group. If for any $a,b\in G$ we have |
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| $a*b=b*a$, we say that the group is \emph{abelian}. Sometimes the expression \emph{commutative group} is used, but this is less frequent. |
$a*b=b*a$, we say that the group is \emph{abelian}. Sometimes the expression \emph{commutative group} is used, but this is less frequent. |
| Abelian groups hold several interesting properties. |
Abelian groups hold several interesting properties. |
| \begin{thm} |
\begin{thm} |
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If $\varphi\colon G\to G$ defined by $ \varphi(x) =x^2$ is an homomorphism, then $G$ is abelian
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If $\varphi:G\to G$ defined by $ \varphi(x) =x^2$ is an homomorphism, then $G$ is abelian
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| \end{thm} |
\end{thm} |
| If such mapping were a homomorphism, we would have |
If such mapping were a homomorphism, we would have |
| \[(xy)^2=\varphi(xy) = \varphi(x)\varphi(y)=x^2y^2\] |
\[(xy)^2=\varphi(xy) = \varphi(x)\varphi(y)=x^2y^2\] |
| that is, $xyxy=xxyy$. Left-mutiplying by $x^{-1}$ and right-multiplying by $y^{-1}$ we are led to |
that is, $xyxy=xxyy$. Left-mutiplying 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. |
| \begin{thm} |
\begin{thm} |
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Any \PMlinkname{subgroup}{Subgroup} of an abelian group is normal.
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Any subgroup of an abelian group is normal.
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| \end{thm} |
\end{thm} |
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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 $G$.
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Let $H$ be a subgroup of the abelian group $G$. Since $ah=ha$ for any $a\inG$ and any $h\in H$ we get $aH=Ha$. That is, $H$ is normal in $G$.
|
| \begin{thm} |
\begin{thm} |
| Quotient groups of abelian groups are also abelian |
Quotient groups of abelian groups are also abelian |
| \end{thm} |
\end{thm} |
| 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. |
