Abstract algebra(Wilkins)

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1、CourseMA2C01,MichaelmasTerm2009Section3:AbstractAlgebraDavidR.WilkinsCopyrightcDavidR.Wilkins2000{2009Contents3AbstractAlgebra323.1BinaryOperationsonSets....................323.2CommutativeBinaryOperations................323.3AssociativeBinaryOperations..................323.4Semigroups.....

2、.......................333.5TheGeneralAssociativeLaw..................343.6Identityelements.........................353.7Monoids..............................353.8Inverses..............................363.9Groups...............................393.10HomomorphismsandIsomorphisms..............

3、.41i3AbstractAlgebra3.1BinaryOperationsonSetsDenitionAbinaryoperationonasetAisanoperationwhich,whenappliedtoanyelementsxandyofthesetA,yieldsanelementxyofA.ExampleThearithmeticoperationsofaddition,subtractionandmultipli-cationarebinaryoperationsonthesetRofrealnumberswhich,whenap-pliedtore

4、alnumbersxandy,yieldtherealnumbersx+y,xyandxyrespectively.Howeverdivisionisnotabinaryoperationonthesetofrealnumbers,sincethequotientx=yisnotdenedwheny=0.(Underabinaryoperationonasetmustdetermineanelementxyofthesetforeverypairofelementsxandyofthatset.)3.2CommutativeBinaryOperationsDenit

5、ionAbinaryoperationonasetAissaidtobecommutativeifxy=yxforallelementsxandyofA.ExampleTheoperationsofadditionandmultiplicationonthesetRofrealnumbersarecommutative,sincex+y=y+xandxy=yxforallrealnumbersxandy.Howevertheoperationofsubtractionisnotcommutative,sincexy6=yxingeneral.(Indeedthe

6、identityxy=yxholdsonlywhenx=y.)3.3AssociativeBinaryOperationsLetbeabinaryoperationonasetA.Givenanythreeelementsx,yandzofasetA,thebinaryoperation,appliedtotheelementsxyandzofA,yieldsanelement(xy)zofA,and,appliedtotheelementsxandyzofA,yieldsanelementx(yz)ofA.DenitionAbinaryoperation

7、onasetAissaidtobeassociativeif(xy)z=x(yz)forallelementsx,yandzofA.ExampleTheoperationsofadditionandmultiplicationonthesetRofrealnumbersareassociative,since(x+y)+z=x+(y+z)and(xy)z=x(yz)forallrealnumbersx,yandz.Howevertheoperationofsubtractionisno