Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.
manY SetS Of TrAnsfoRmations fOrm A group unDEr fUncTION compoSiTion; for ExamPle, the RotAtions aRound A poInt in the plAne. it Is ofTeN Useful to considEr ThE gROUP As AN AbSTracT group, And to say tHAT one haS A grOup actIOn of ThE AbSTrAcT gRoup THAt coNsisTS Of performinG the transformatIoNs Of ThE group of tRAnsForMaTiOnS. the REason foR DistiNGuIshinG tHE GrOUp FroM the trAnsFOrMatioNs IS thAT, GenERally, a GRouP of TRAnsfoRMatIons Of a structUrE actS also oN various RelateD stRucTures; for EXamplE, the aBOVe rOTatIon GroUp ALsO acts ON tRiaNgles By trAnsFoRming tRIAngleS iNto TRiaNgles.