2385. ACTIONS OF GROUPSExample 5.1.4.Any groupGacts on itself by left multiplication. Thatis, forg2Gandx2G,gxis just the usual product ofgandxinG. The homomorphism, or associative, property of the action is just theassociative law ofG. There is only one orbit. The action ofGon itself byleft multiplication is often called theleft regular action.Definition 5.1.5.An action ofGonXis calledtransitiveif there is onlyone orbit.That is, for any two elementsx; x02X, there is ag2Gsuch thatgxDx0. A subgroup of Sym.X/is calledtransitiveif it actstransitively onX.Example 5.1.6.LetGbe any group andHany subgroup. ThenGactson the setG=Hof left cosets ofHinGby left multiplication,g.aH/D.ga/H. The action is transitive.Example 5.1.7.Any groupGacts on itself by conjugation: Forg2G,definecg2Aut.G/Sym.G/bycg.x/Dgxg1. It was shown inExercise2.7.6that the mapg7!cgis a homomorphism. The orbits ofthis action are called theconjugacy classesofG; two elementsxandyareconjugateif there is ag2Gsuch thatgxg1Dy. For example,it was shown in Exercise2.4.14that two elements of the symmetric group
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