> < ^ Date: Tue, 06 Mar 2001 10:21:31 -0500 (EST)
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
< ^ Subject: Re: Lattice of subgroups

Dear GAP-Forum,

Bj"orn Vandenberghe asked:

> We have a problem with the lattice of subgroups of a perfect group. We =
(I suppose this is a theory question. If you actually found a bug in
the lattice calculation -- which I hope would not happen -- please send us
the example to gap-trouble@dcs.st-and.ac.uk.)

thought that the lattice was found with cyclic extension methods in Gap. =
Now we have found several examples which in our opinion can not be =
found by cyclic extension. Our question is now: thus Gap work with =
cyclic extension to find de lattice of a perfect groups, (or to find the =
conjugacyrepresentatives) ??

You are right in that a pure cyclic extension will never find perfect
subgroups. Therefore the lattice computation first obtains (representatives
of) all perfect subgroups (command: RepresentativesPerfectSubgroups') and
extends these as well.

To find these perfect subgroups, one can show, that every perfect
subgroup P of G must lie in the perfect residuum G^\infty (the subgroup where
the derived series becomes stationary).

GAP then uses (essentially) the operation IsomorphicSubgroups' to check for
all groups in the perfect groups library (which are candidates for embedding
by size and element orders) whether they actually embed.

(Since the perfect groups library is finite (and contains some gaps) it is
not guaranteed that GAP will be able to compute the subgroup lattice of a
group of size >= 245760. However at this size, usually memory problems arise
already earlier so that this problem has not yet arisen in prcatice.)

If you set

SetInfoLevel(InfoLattice,n);

with n=1 or n=2 before calling a lattice computation, GAP will print
information about this process. You can also find the code for cyclic
extension in in lib/grplatt.gi'.

Finally, let me mention that GAP 4 will compute the lattice of a group which
is known to be solvable by another (more efficient) method elementary
abelian extension'' which is described in:

@article{hulpke99,
author =	 "Alexander Hulpke",
title =	 "Computing Subgroups Invariant Under a Set of Automorphisms",
journal =	 "J.Symb.Comp."
pages =	 "415--427",
volume =	 27,
number =	 4,
note =	 "(ID jsco.1998.0260)",
year =	 "1999"
}
`

Best wishes,

Alexander Hulpke

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