> < ^ Date: Tue, 23 Aug 1994 22:09:00 +1000
> < ^ From: Eamonn O'Brien <obrien@math.auckland.ac.nz >
< ^ Subject: some group theoretic questions

I write in response to the following:

From: Bruce Kaskel <kaskel@math.berkeley.edu>

Please direct e-mail responses to Ruvain Gittelman at:

1) What is the largest integer n such that ALL groups of order less than n
   are known? Where is this information to be found?
2) What is the largest n such that all groups of order 2  are known?
   Same for odd primes.

3) What is known about the automorphism group towers of finite groups
   which may have nontrivial center? Are groups whose tower terminates
   in {e} known?

1. I believe that all groups of orders at most 256 are determined, apart
from those of order 192.

Presentations for the groups of order at most 100 are distributed with each
of GAP and Magma. These are distilled from various sources, two primary
ones being:

Joachim Neub\"{u}ser (1967), ``Die Untergruppenverb\"{a}nde der Gruppen der
Ordnungen $\leq 100$ mit Ausnahme der Ordnungen $64$ und $96$",
Habilitationsschrift, Kiel.

R. Laue (1982), ``Zur Konstruktion und Klassifikation endlicher
aufl\"{o}sbarer Gruppen". Bayreuth. Math. Schr. No. 9.

In addition, presentation for the groups of orders dividing 256 and 729 are
supplied as part of the group libraries of both systems.

Groups whose order involves certain numbers of prime factors are known in
various cases: for example, the groups of order p^2*q, where p and q are
distinct primes.

In the introduction to a 1990 paper of James, Newman & O'Brien [see full
reference details below], we mention a few references which you may wish to
follow up. Also Mark Short (at Murdoch University, Western Australia) and I
prepared a bibliography in the late 1980s on the history of group
determinations. I am happy to supply it to you on request.

2. The 2-groups of order dividing 256 and the 3-groups of order dividing 729
have been completely determined (and, as mentioned above, are supplied as
part of the GAP library). Details of the results and methods can be found in
the following papers:

Marshall Hall, Jr., & James K. Senior (1964), The Groups of
Order $2^n$ (n$\leq6$). Macmillan, New York.

Rodney James, M.F. Newman & E.A. O'Brien (1990), ``The groups of
order 128",  J. Algebra 129, 136-158.

E.A. O'Brien (1991), ``The groups of order 256",
J. Algebra  143, 219-235.

E.A. O'Brien (1990), ``The p-group generation
algorithm", J. Symbolic Comput.  9, 677-698.

Various results on the groups of order p^6 (for odd p) are in the
literature. A modern determination of the groups of order dividing p^6 is
presented in the following paper:

Rodney James (1980), ``The Groups of Order $p^6$ ($p$ an Odd Prime)",
Math. Comp. 34, 613-637.

This list seems reliable for the groups of order dividing p^5. However, M.F.
Newman and I are aware of a number of errors in this list for groups of
order p^6 and plan -- at some point -- to prepare a corrected version.
Again, we are happy to supply our corrections on request, but we make no
claims on its completeness.

A determination of the groups of exponent p and order p^7 can be found in a
paper by Wilkinson which appeared in the J. Algebra in 1988. [I don't have
full reference details handy.]

Of course, in some sense, the computational tools to carry out the
determination of particular classes of p-groups are now available -- via the
p-group generation algorithm [see O'Brien, 1990, above for full details].
The implementation of this algorithm is available as a package in GAP and,
while limited in the range of application, it can be used in certain cases
to determine certain classes of p-groups. For example, in its current form,
it could be used to determine most of the approximately *10 million* groups
of order 512, or most of the groups of order 3^7. However, one might wish
to think carefully before embarking on either project.

Eamonn O'Brien

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