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From: hen@bu-cs.UUCP (Bill Henneman)
Newsgroups: net.math
Subject: Re: Finding groups of order N
Message-ID: <765@bu-cs.UUCP>
Date: Thu, 14-Nov-85 08:32:43 EST
Article-I.D.: bu-cs.765
Posted: Thu Nov 14 08:32:43 1985
Date-Received: Fri, 15-Nov-85 20:35:46 EST
References: <1208@bbncc5.UUCP>
Organization: Boston Univ Comp. Sci.
Lines: 25

I spent some time in the early 70's writing a program which helped
enumerate the solvable groups of order N.  The only practical
group-theoretic tricks I could make use of were:

 Factoring N and using the Sylow theorems can give an upper bound on the
possible number of groups of order N: when you have found this many,
you're done.  For example, N=15 has only 1 group, no search necessary.
If you're allowing non-abelian simple groups, I don't know how to get
the upper bound, but it can probably be derived.

 The Sylow theorems also tell you something about the structure of the
subgroup lattice, and something about the normal subgroup lattice: you
can build all the groups N= 21 as extensions of groups of order 7 (since
7k+1 divides 21 only when k=0).

 The Todd-Coxeter algorithm can be used to generate candidate extensions
of A by B, but you still have to do an isomorphism check on the
candidates.

 Keeping the spectrum (a list of indices of the elements of the group)
can cut the search space for possible isomorphic groups.  For the
special case of enumerating groups of prime-power order, this gives a
great saving.
 
 That's all I know.  Hope it helps.