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 2n for n leq6, Neubüser listed the groups of order at most 100 except 64 and 96 and Laue added the groups of order 96, see HS64, Neu67 and 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 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 2n for n leq8 and the groups of order 3n for n leq6, see OBr88, and the resulting groups are available in GAP. Moreover, the large number of groups of order 28 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 BE99. These methods have been used to construct the non-nilpotent groups of order at most 1000, see 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 thumb 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 = 26 cdot3 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 = 28 cdot3 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 = 52 cdot7 cdot31 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 BE99 and BE1000.
Comments and suggestions on this share package are very welcome. Please send them to
centerline firstname.lastname@example.org or email@example.com.
Bug reports should also be e-mailed to either of these addresses.
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