Full group

In mathematics , especially in group theory , a group is called complete if its center consists only of the neutral element and every automorphism is internal .

Examples

• The symmetric groups are complete, except when n is 2 or 6. In the case n = 2, the center does not only consist of the neutral element and has an outer automorphism. Taking Cayley's theorem into account , it follows that every finite group can be embedded in a complete group. ${\displaystyle S_{n}}$ ${\displaystyle S_{6}}$

• One can show that the automorphism group of a non-Abelian , simple group is complete.

properties

• If a group G is complete, then the homomorphism of G into the automorphism group is an isomorphism . It is injective because the center consists only of the neutral element and it is surjective because every automorphism is internal. ${\displaystyle g\mapsto \mathrm {Int} (g):x\mapsto gxg^{-1}}$ ${\displaystyle \mathrm {Aut} (G)}$

• From the above statement it follows that a complete group is always isomorphic to its automorphism group.

• The reverse of the above statement is not true, i.e. a group can be isomorphic to its automorphism group without being complete. We show that this is the case for the 8-element dihedral group . To this end, let G and H be two isomorphic groups, an element of order 4, and b an element of G that is not in the subgroup generated by a ; so be an element of order 4 and not in the subgroup generated by c . Then there is exactly one isomorphism that maps a to c and b to c . Applying this to G = H = D ₄ , one can see that there are exactly 8 automorphisms on . More precisely, one finds that isomorphic too . Ultimately, however, a 2-group cannot be complete, because the center does not only consist of the neutral element, it contains an element of order 2. ${\displaystyle D_{4}}$ ${\displaystyle D_{4}}$ ${\displaystyle a\in G}$ ${\displaystyle c\in H}$ ${\displaystyle d\in H}$ ${\displaystyle G\rightarrow H}$ ${\displaystyle D_{4}}$ ${\displaystyle \mathrm {Aut} (D_{4})}$ ${\displaystyle D_{4}}$ ${\displaystyle D_{4}}$

• If a complete group K is normal divisor in a group G , then G is the direct product of K and the centralizer of K in G. ${\displaystyle \ C_{G}(K)}$

Proof: From the fact that K normal subgroup in G , it follows that even a normal subgroup in G is. Furthermore is the center of K and therefore consists only of the neutral element. Now be arbitrary. Since K is normal, the inner automorphism induced by G an automorphism on K . Since K is complete, this automorphism of K is internal, so there is an element h in K with for all . So is off , that means g belongs to and therefore . ${\displaystyle \ C_{G}(K)}$ ${\displaystyle \ K\cap C_{G}(K)}$ ${\displaystyle g\in G}$ ${\displaystyle \ x\mapsto gxg^{-1}}$ ${\displaystyle \ gkg^{-1}=hkh^{-1}}$ ${\displaystyle k\in K}$ ${\displaystyle \ h^{-1}g}$ ${\displaystyle \ C_{G}(K)}$ ${\displaystyle \ KC_{G}(K)}$ ${\displaystyle \ G=KC_{G}(K)}$

• From what has just been said, it follows that a complete group is a direct factor in any group in which it is included as a normal divisor. This property characterizes the complete groups: If a group K is a direct factor in every group in which it is contained as a normal divisor, then K is complete. This is shown very easily using the fact that group K is a direct factor of its holomorph under the given conditions .

• One can show that if a group G has a trivial center, this also applies to its automorphism group . The canonical homomorphisms , etc. are therefore injective and you can the ascending sequence of groups ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle G\rightarrow \mathrm {Aut} (G)}$ ${\displaystyle \mathrm {Aut} (G)\rightarrow \mathrm {Aut} (\mathrm {Aut} (G))}$

${\displaystyle G\leq \mathrm {Aut} (G)\leq \mathrm {Aut} (\mathrm {Aut} (G))\leq \ldots }$

consider that one of the Automorphismenturm of G called. Helmut Wielandt showed in 1939 that for a finite group with a trivial center, its automorphism tower becomes stationary, that is, a complete group is obtained.

• For infinite groups with a trivial center it can be shown that the automorphism tower defined above does not necessarily become stationary, in other words that one does not necessarily get a complete group. However, one can define a family for any non-empty ordinal number as follows . It is , if a direct predecessor has, and sets the same as the inductive limit of wherein the elements passes through, as strictly less are. Simon Thomas showed in 1985 that there is an ordinal number for every (finite or infinite) group with a trivial center, so that the corresponding transfinite automorphism tower becomes stationary. ${\displaystyle \ \Omega }$ ${\displaystyle \ (G_{\omega })_{\omega \in \Omega }}$ ${\displaystyle \ G_{0}=G}$ ${\displaystyle G_{\omega }:=\mathrm {Aut} (G_{\lambda })}$ ${\displaystyle \omega >0}$ ${\displaystyle \lambda }$ ${\displaystyle G_{\omega }}$ ${\displaystyle \ G_{\lambda }}$ ${\displaystyle \ \lambda }$ ${\displaystyle \ \Omega }$ ${\displaystyle \omega }$

Individual evidence

1. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, page 158
2. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, page 160
3. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, page 162
4. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, exercise 7.15 on page 167
5. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, proof of theorem 7.15, page 163
6. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, p. 164
7. ^ H. Wielandt: A generalization of the invariant subgroups , Mathematische Zeitschrift 1939, pages 209-244, see also IM Isaacs, Finite Group Theory , American Mathematical Society 2008, pages 278-284
8. ↑ Simon Thomas, The automorphism tower problem , Proceedings of the American Mathematical Society (1985), Volume 95, pages 166-168