238 5. ACTIONS OF GROUPS Example 5.1.4. Any group G acts on itself by left multiplication. That is, for g 2 G and x 2 G , gx is just the usual product of g and x in G . The homomorphism, or associative, property of the action is just the associative law of G . There is only one orbit. The action of G on itself by left multiplication is often called the left regular action . Deﬁnition 5.1.5. An action of G on X is called transitive if there is only one orbit. That is, for any two elements x;x0 2 X , there is a g 2 G such that gx D x0 . A subgroup of Sym .X/ is called transitive if it acts transitively on X . Example 5.1.6. Let G be any group and H any subgroup. Then G acts on the set G=H of left cosets of H in G by left multiplication, g.aH/ D .ga/H . The action is transitive. Example 5.1.7. Any group G acts on itself by conjugation: For g 2 G , deﬁne c g 2 Aut .G/ ± Sym .G/ by c g .x/ D gxg ± 1 . It was shown in Exercise 2.7.6 that the map g 7! c g is a homomorphism. The orbits of this action are called the
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