TITLE: classification of small complete groups
QUESTION [6 upvotes]: I take it there isn't a classification of finite complete groups yet. Has someone put together a classification of small complete groups? I.e. $S_4$, $\text{Aut}(G)$ for $G$ simple, $\text{Hol}(C_p)$ for $p$ odd prime are complete. The smallest complete group not of one of these forms is $H$ of order $\left|H\right|$, which generalises to a class $P$. The smallest complete group not of one of these forms is $J$, and so on, until we run out of ideas.

REPLY [2 votes]: This is not a classification, but it is interesting to notice that every finite complete group can be obtained by the following process: start with any finite centerless group $G$ and iterate the automorphism group operations to build the automorphism tower:

$G\to\mathop{Aut}(G)\to \mathop{Aut}(\mathop{Aut}(G))\to\cdots$

Each group maps into the next by the inner automorphisms. If $G$ is centerless, then so is $\mathop{Aut}(G)$ and all the groups in the tower are centerless. The tower terminates at a complete group, a centerless group that is isomorphic to $\mathop{Aut}(G)$ by the canonical map. 
It is a beautiful theorem of Wielandt (1939) that every finite centerless group has an automorphism tower that terminates in finitely many steps. Thus, every finite complete group arises in this fashion. 
This question is also considered at this classic MO question. 
Truly interesting things occur if one is willing to push things harder by continuing the iteration transfinitely. That is, having built the finite part of the automorphism tower, which has an associated system of mappings, one may simply take the direct limit to produce the limit group $G_\omega$ at stage $\omega$ and continue the iteration transfinitely with 

$G_0\to G_1\to G_2\to\cdots G_\omega\to G_{\omega+1}\to\cdots\to G_\alpha\to\cdots$

where one takes the automorphism group at successor ordinal stages and direct limits of the system at limit ordinal stages. Simon Thomas proved the wonderful theorem that every centerless group leads in this transfinite manner to a complete group, where the process stops (see Proceedings AMS article). I extended this result by proving:
Theorem. Every group $G$ has a terminating transfinite automorphism tower. 
The proof proceeds by showing that every $G$ leads eventually to a centerless group in the transfinite tower, whose subsequent tower then terminates by Thomas' theorem.  You can see my paper at the ArXiv, published in the Proceedings of the AMS. 
It is an open question how tall the automorphism tower of a finite group can be. Examples are known with nontrivial centers that do not terminate at any finite stage and make it out to $\omega+n$ for any desired $n$, but the smallest upper bound is on the order of the least inaccessible cardinal. 
Simon Thomas is writing a book on the subject of the automorphism tower problem, and the preliminary versions that I have seen are excellent.