Unformatted text preview: 248 5. ACTIONS OF GROUPS concerned. For example, a rotational symmetry of the cube moves the individual points of the cube around but preserves the structure of the cube. A structure preserving bijection of any sort of object can be regarded as a symmetry of the object, and the set of symmetries always constitutes a group. If, for example, the structure under consideration is a group, then a structure preserving bijection is a group automorphism. In this section, we will work out a few examples of automorphism groups of groups. Recall that the inner automorphisms of a group are those of the form c g .x/ D gxg 1 for some g in the group. The map g 7! c g is a ho- momorphism of G into Aut .G/ with image the subgroup Int .G/ of inner automorphisms. The kernel of this homomorphism is the center of the group and, therefore, Int .G/ Š G=Z.G/ . Observe that the group of inner automorphisms of an abelian group is trivial, since G D Z.G/ ....
View Full Document
- Fall '08
- Algebra, Abelian group, Isomorphism, Group isomorphism, Group homomorphism, automorphism