Sei
die Gruppe aus Bsp. 398). Man bestimme die vom Element
erzeugte Untergruppe sowie deren Nebenklassen in
.
![{\displaystyle \Gamma _{16}=\{{\overline {1}},{\overline {3}},{\overline {5}},{\overline {7}},{\overline {9}},{\overline {11}},{\overline {13}},{\overline {15}}\}}](/index.php?title=Spezial:MathShowImage&hash=a923f590d6bc1753f657cc2fdd9f37f8&mode=mathml)
![{\displaystyle U=\{{\overline {1}},{\overline {9}}\}}](/index.php?title=Spezial:MathShowImage&hash=ccb83c747d25a1a63d1dccb5d80e2d8b&mode=mathml)
ist assoziativ, da die Assoziativität in ganz
gilt.
![{\displaystyle {\overline {1}}*{\overline {1}}={\overline {1}}}](/index.php?title=Spezial:MathShowImage&hash=2d0693cdb056df3d0f3fe97a7a5d00bc&mode=mathml)
![{\displaystyle {\overline {1}}*{\overline {9}}={\overline {9}}}](/index.php?title=Spezial:MathShowImage&hash=961bca09c8df2e863e49dd6d03d77b41&mode=mathml)
![{\displaystyle {\overline {9}}*{\overline {1}}={\overline {9}}}](/index.php?title=Spezial:MathShowImage&hash=20cf4f7526fac70811a10573b61b11aa&mode=mathml)
![{\displaystyle {\overline {9}}*{\overline {9}}={\overline {1}}}](/index.php?title=Spezial:MathShowImage&hash=bc4ef6c69b5c390b7d975139a7270d6a&mode=mathml)
![{\displaystyle \Rightarrow abgeschlossen}](/index.php?title=Spezial:MathShowImage&hash=fbbbc17a6f6eeee1b669c519054fafa0&mode=mathml)
![{\displaystyle {\overline {1}}*{\overline {1}}={\overline {1}}}](/index.php?title=Spezial:MathShowImage&hash=2d0693cdb056df3d0f3fe97a7a5d00bc&mode=mathml)
![{\displaystyle {\overline {1}}*{\overline {9}}={\overline {9}}}](/index.php?title=Spezial:MathShowImage&hash=961bca09c8df2e863e49dd6d03d77b41&mode=mathml)
![{\displaystyle e={\overline {1}}\Rightarrow neutrales\,Element}](/index.php?title=Spezial:MathShowImage&hash=903e75fb1f935cc96d8db80c35021b2c&mode=mathml)
![{\displaystyle ({\overline {1}})^{-1}={\overline {1}}}](/index.php?title=Spezial:MathShowImage&hash=75e96ea6020707baf4cfcf97c20fb34a&mode=mathml)
![{\displaystyle ({\overline {9}})^{-1}={\overline {9}}}](/index.php?title=Spezial:MathShowImage&hash=0e68de87182c66dd72c17d1ae12bf0bf&mode=mathml)
![{\displaystyle \Rightarrow inverse\,Elemente\,existieren}](/index.php?title=Spezial:MathShowImage&hash=cbdb11dc83646252da75ab45bf9f437e&mode=mathml)
Damit ist bewiesen, dass
eine Gruppe ist, welche von
erzeugt wurde.
![{\displaystyle {\overline {1}}\circ U={\overline {9}}\circ U=\{{\overline {1}},{\overline {9}}\}}](/index.php?title=Spezial:MathShowImage&hash=9eb40645543803f4a382eea2c9c354c6&mode=mathml)
![{\displaystyle {\overline {3}}\circ U={\overline {11}}\circ U=\{{\overline {11}},{\overline {3}}\}}](/index.php?title=Spezial:MathShowImage&hash=90573f5fc8bf67a42fea30eca79680e7&mode=mathml)
![{\displaystyle {\overline {5}}\circ U={\overline {13}}\circ U=\{{\overline {13}},{\overline {5}}\}}](/index.php?title=Spezial:MathShowImage&hash=bdf09dabdf7b91400b14d61335f38825&mode=mathml)
![{\displaystyle {\overline {7}}\circ U={\overline {15}}\circ U=\{{\overline {7}},{\overline {15}}\}}](/index.php?title=Spezial:MathShowImage&hash=401ded8102e4847bd4f01cd7c60efc6b&mode=mathml)
![{\displaystyle L=\{\{{\overline {1}},{\overline {9}}\},\{{\overline {3}},{\overline {11}}\},\{{\overline {5}},{\overline {13}}\},\{{\overline {7}},{\overline {15}}\}}](/index.php?title=Spezial:MathShowImage&hash=4980f7de2d9b51227c6aab1b164e24a9&mode=mathml)
Da die Restklassenmultiplikation kommumativ ist, entspricht die Linksnebenklasse der Rechtsnebenklasse.
ist Normalteiler von