Sei
die Gruppe aus Bsp. 399). Man bestimme die vom Element
erzeugte Untergruppe sowie deren Nebenklassen in
.
![{\displaystyle \Gamma _{18}=\{{\overline {1}},{\overline {5}},{\overline {7}},{\overline {11}},{\overline {13}},{\overline {17}}\}}](/index.php?title=Spezial:MathShowImage&hash=f41b0d809a427084903d41fd9c42ff76&mode=mathml)
![{\displaystyle U=\{{\overline {7}},{\overline {13}},{\overline {1}}\}}](/index.php?title=Spezial:MathShowImage&hash=43842807560fec3112090ee2edf286aa&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 {7}}={\overline {7}}}](/index.php?title=Spezial:MathShowImage&hash=adc40de40ef3cfa9a52f9c0f4f51c7a8&mode=mathml)
![{\displaystyle {\overline {1}}*{\overline {13}}={\overline {13}}}](/index.php?title=Spezial:MathShowImage&hash=599a2dfe6f3e95a3e099a4d032f6e47c&mode=mathml)
![{\displaystyle {\overline {7}}*{\overline {13}}={\overline {1}}}](/index.php?title=Spezial:MathShowImage&hash=e9784b11a4a0e98011b5cd13a825a340&mode=mathml)
![{\displaystyle {\overline {7}}*{\overline {7}}={\overline {13}}}](/index.php?title=Spezial:MathShowImage&hash=8d14e5821417d6a9d5b947b36598266b&mode=mathml)
![{\displaystyle {\overline {13}}*{\overline {13}}={\overline {7}}}](/index.php?title=Spezial:MathShowImage&hash=3b9c855b4a9c5b6ff265a419a3a3837b&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 {7}}={\overline {7}}}](/index.php?title=Spezial:MathShowImage&hash=adc40de40ef3cfa9a52f9c0f4f51c7a8&mode=mathml)
![{\displaystyle {\overline {1}}*{\overline {13}}={\overline {13}}}](/index.php?title=Spezial:MathShowImage&hash=599a2dfe6f3e95a3e099a4d032f6e47c&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 {7}})^{-1}={\overline {13}}}](/index.php?title=Spezial:MathShowImage&hash=c952510af978a45b1493f36013db8659&mode=mathml)
![{\displaystyle ({\overline {13}})^{-1}={\overline {7}}}](/index.php?title=Spezial:MathShowImage&hash=6ba988acd419f3a2c5cee0b07d2a00e0&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 {7}}\circ U={\overline {13}}\circ U=\{{\overline {1}},{\overline {7}}\,{\overline {13}}\}}](/index.php?title=Spezial:MathShowImage&hash=a38f7ca08bbc7ae7f5fa96748da4a22f&mode=mathml)
![{\displaystyle {\overline {5}}\circ U={\overline {11}}\circ U={\overline {13}}\circ U=\{{\overline {5}},{\overline {11}},{\overline {13}}\}}](/index.php?title=Spezial:MathShowImage&hash=2c9e2d2cc25b29fc4fb5042e94e051a3&mode=mathml)
![{\displaystyle L=\{\{{\overline {1}},{\overline {7}}\},\{{\overline {13}}\},\{{\overline {5}},{\overline {11}}\},\{{\overline {17}}\}}](/index.php?title=Spezial:MathShowImage&hash=b737c1b4f01a4538994f929a4b6ec14d&mode=mathml)
Da die Restklassenmultiplikation kommumativ ist, entspricht die Linksnebenklasse der Rechtsnebenklasse.
ist Normalteiler von