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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.

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"escalated" to <a href="mathoverflow.net/questions/39315/…; –  baby monster Sep 19 '10 at 16:57

2 Answers 2

I'm not sure you've asked a question with an answer, but I'll try to address what I can. Surely people have looked at small complete groups, but people do not tend to publish such things.

The smallest complete group that is not S4, Aut(simple), nor Hol(p) is S3 x Hol(5). Direct products of complete groups are often but certainly not always complete. You might try to find reasonable conditions on when the direct product of completes is complete. Other people have written down some conditions, but I can't recall where.

An infinite family omitted are the AΓL(1,q) of semi-linear transformations of a one-dimensional vector space. If q is prime, this is just Hol(q).

The next two groups not covered might be interesting to understand. They are of the form G⋉V where G is complete and V is an absolutely irreducible G-module (G being AΓL(1,9) or Hol(5), V being smallest non-trivial dimension and rational). I don't think all such groups are complete, but it would be interesting to know conditions on G,V to make them complete.

I don't think one will get any good sort of classification as there are just weird complete groups. One group V of order 128 has an automorphism G of order 7 (and several other outer automorphisms), but the semi-direct product G⋉V of order 128*7 is complete. (SmallGroup(432,520) and SG(1344,816) are almost as weird). I at least find it a little disturbing to have small subgroups of Hol(V) be complete when neither V nor Hol(V) is complete. The other G⋉V examples are like this too, but perhaps I find it less disturbing when V is abelian.

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Thanks for this. On MO they suggested I should probably just download GAP and figure it out. I have downloaded it and am just getting used to it. It looks like this would get me to order 2000. Now e.g. Hol(3) X Hol(5) is already order 120. So the prospect of finding sufficient conditions on when the direct product of two complete groups is complete that works up to order 2000 with maybe a couple of exceptions doesn't seem so bad. –  baby monster Sep 26 '10 at 22:31
    
Say .. using GAP I am unable to find any complete group of order 896 = 128*7. What is the group V of order 128 you are talking about? –  baby monster Sep 29 '10 at 2:14
    
Sorry, the 128*7 group is not centerless, just outerless. SmallGroup(896,564). It has Inn(G) = Aut(G), but G ≠ Inn(G) and Aut(G) ≠ Aut(Aut(G)). –  Jack Schmidt Sep 30 '10 at 15:08

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.

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