Introduction
Themappingofacollectiontoitselfiscalledatransformation.Asamapping,twotransformscanbemultiplied.Agroupofseveral-transformationsunderthismultiplicationiscalledatransformationgroup.
Transformationgroup
[transformationgroup]
People'sunderstandingofgroupsbeginswithtransformationgroups.
E.Galois(E.Galois)Theruleofjudgingtheexistenceoftheradicalsolutionofapolynomialequationisthroughtheautomorphismgroup(transformationgrouponthetop)oftheresearchdomainandthetransformationgrouponthesetofrootsAndgotit.Therefore,peopleusuallyregardGaloisasthefounderofgrouptheory.
A.Cayleyputforwardtheconceptofabstractgroup,andthentherewasresearchonabstractgroup.Abasicfactisthatanygroupisisomorphictoatransformationgroup.Infact,ifGisagroupand,thentherightmultiplicationtransform,whichmapsintoaproduct,whichisasetAtransformationonG.WhengtraversesalltheelementsinG,theentirerightmultiplicationtransformationbecomesaunderthetransformationmultiplication.ItisthetransformationgrouponthesetG,andthemappingistheisomorphismofthegroupGtothetransformationgroup.