Complete group
In mathematics, a group Template:Math is said to be complete if every automorphism of Template:Math is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center.
Equivalently, a group is complete if the conjugation map, Template:Math (sending an element Template:Math to conjugation by Template:Math), is an isomorphism: injectivity implies that only conjugation by the identity element is the identity automorphism, meaning the group is centerless, while surjectivity implies it has no outer automorphisms.
ExamplesEdit
As an example, all the symmetric groups, Template:Math, are complete except when Template:Math}. For the case Template:Math, the group has a non-trivial center, while for the case Template:Math, there is an outer automorphism.
The automorphism group of a simple group is an almost simple group; for a non-abelian simple group Template:Math, the automorphism group of Template:Math is complete.
PropertiesEdit
A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the converse need not hold: for example, the dihedral group of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see Template:Harv.
Extensions of complete groupsEdit
Assume that a group Template:Math is a group extension given as a short exact sequence of groups
with kernel, Template:Math, and quotient, Template:Math. If the kernel, Template:Math, is a complete group then the extension splits: Template:Math is isomorphic to the direct product, Template:Math. A proof using homomorphisms and exact sequences can be given in a natural way: The action of Template:Math (by conjugation) on the normal subgroup, Template:Math, gives rise to a group homomorphism, Template:Math. Since Template:Math and Template:Math has trivial center the homomorphism Template:Math is surjective and has an obvious section given by the inclusion of Template:Math in Template:Math. The kernel of Template:Math is the centralizer Template:Math of Template:Math in Template:Math, and so Template:Math is at least a semidirect product, Template:Math, but the action of Template:Math on Template:Math is trivial, and so the product is direct.
This can be restated in terms of elements and internal conditions: If Template:Math is a normal, complete subgroup of a group Template:Math, then Template:Math is a direct product. The proof follows directly from the definition: Template:Math is centerless giving Template:Math is trivial. If Template:Math is an element of Template:Math then it induces an automorphism of Template:Math by conjugation, but Template:Math and this conjugation must be equal to conjugation by some element Template:Math of Template:Math. Then conjugation by Template:Math is the identity on Template:Math and so Template:Math is in Template:Math and every element, Template:Math, of Template:Math is a product Template:Math in Template:Math.
ReferencesEdit
- Template:Citation
- Template:Citation (chapter 7, in particular theorems 7.15 and 7.17).