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%W  grpconst.tex          GrpConst documentation             Bettina Eick
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\Chapter{Preface}

The determination of all groups of a given order up to isomorphism
is a central problem in finite group theory. It has been initiated
in 1854 by A. Cayley who constructed the groups of order 4 and 6.

A large number of publications followed Cayley's work. For example,
Hall and Senior determined the groups of order $2^n$ for $n \leq 6$,
Neub{\accent127 u}ser listed the groups of order at most 100 except 64 and 96
and Laue added the groups of order 96, see \cite{HS64}, \cite{Neu67}
and \cite{Lau82}. These determinations partially relied on the
help of computers, but a general algorithm to construct groups had
not been used. The resulting catalogue of groups of order at most
100 has been available in GAP 3.

Then Newman and O'Brien introduced an algorithm to determine
groups of prime-power order, see \cite{OBr90}. An implementation 
of this method is available in the ANUPQ share package of GAP. 
This method has been used to compute the groups of order $2^n$ for 
$n \leq 8$ and the groups of order $3^n$ for $n \leq 6$, see
\cite{OBr88}, and the resulting groups are available in GAP.
Moreover, the large number of groups of order $2^8$ shows that
algorithmic methods are the only sensible way for group determinations
in this range.

In this share package we introduce practical methods to determine
up to isomorphism all groups of a given order. The algorithms 
are described in \cite{BE99}. These methods have been used to 
construct the non-nilpotent groups of order at most 1000, see 
\cite{BE1000}. The resulting catalogue of groups is available within
the small groups library of GAP 4. 

Our methods are not limited to groups of order at most 1000
and thus may be used to determine all or certain groups of 
higher order as well. However, it is not easy to say for 
which orders our methods are still practical and for which
not. As a rule of thump one can say that the number of 
primes and the size of the prime-powers contained in the 
factorisation of the given order determine the practicability 
of the algorithm; that is, the more primes are contained in
the factorisation the more difficult the determination gets. 

As an example, the construction of all non-nilpotent groups of order 
$192 = 2^6 \cdot 3$ takes 17 minutes on an PC 400 Mhz. This is a medium 
sized application of our methods.  However, the construction of the groups
of order $768 = 2^8 \cdot 3$ takes already rather long (a few days) and 
can be considered as a limit of our methods. On the other hand, the 
groups of order $5425 = 5^2 \cdot 7 \cdot 31$ can be determined in 5 sec.
Moreover, if the determination of groups is restricted to groups
with certain properties, then this might increase the efficiency
of the construction process considerably. We include some example
applications of our methods to illustrate this at the end of the
manual.

Finally, we mention that the correctness of our algorithms is very
hard to check for a user; in particular, since there are no other
algorithms for the same purpose available, it might be difficult to
verify that our methods compute all desired groups. Thus we note here 
that methods implemented in this share package have been used to compute 
large parts of the Small Groups library and this, in turn, has been 
checked by the authors as described in \cite{BE99} and \cite{BE1000}.

Comments and suggestions on this share package are very welcome.
Please send them to 

\centerline{ eick@mathematik.uni-kassel.de or 
             Hans-Ulrich.Besche@math.rwth-aachen.de.}

Bug reports should also be e-mailed to either of these addresses.

