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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.

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