> < ^ Date: Fri, 15 Mar 1996 14:38:08 +1553
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
^ Subject: Re: Subgroup Lattices (Untergruppengraphen)

Dear GAP-Forum,

Katharina Geissler wrote: (translation by me)

I'm searching a function for the transitive groups of degree 12 to display
how the several subgroups of S12 are contained within each other.

What you're asking for are actually two things. The second is a graphical
display. In principle you could persuade 'GraphicLattice' to display a
partial lattice for the transitive groups. See the ongoing discussion about
XGAP and its possible extensions. We are currently discussing several
possibilities, but so far nothing specific has been planned.
Nevertheless I doubt that this is what you really want, unless you have a
BIG monitor: there are 301 transitive groups of degree 12. Displaying all
containments of conjugates will probably result in a mess of lines on the
screen which is not really usable. Anyhow, if you're using pen and paper you
could still try to produce such a picture based on the information you would
provide to 'GraphicLattice'. (BTW.: If you finally get such a picture I
would be VERY interested to get a copy of it. So far I've been too lazy
to do it myself.)

The first (and bigger) problem is to get the actual containment information.
So far there is no function to compute this with a single command in GAP.
I will describe, however, how you can compute it yourself, but it might
take you some time to do so:

The process will yield not only information *whether* a group is contained
in another, but also information how many conjugacy classes exist if the
subgroup is maximal. (You will need this information if you want to identify
Galois groups, as I suppose. You should note as well, that GAP already
contains information about resolvents distinguishing the groups. Probably
this is of help. Write to me directly if you want more information about this.)

If you can compute representatives of the conjugacy classes of maximal
subgroups of each transitive group, you are done. Non-maximal containment
simply follows by induction. 265 of the transitive groups are solvable. By
converting them to an AgGroup and then to a SpecialAgGroup, you can compute
representatives of the conjugacy classes of maximal subgroups and
transfer them (using the components .bijection in the SAgGroup and the
AgGroup) back in the permutation group. There, you select the ones which are
transitive. The command 'TransitiveIdentification' then tells you for each
of the representatives the number in the list of transitive groups, avoiding
conjugacy tests in S12.

This leaves 36 non-solvable groups. Coping with S12 and A12 is quite simple,
as the maximal subgroups are classified already (the imprimitive ones are
wreath products, the primitive ones are dealt with in the ATLAS).
8 of the remaining groups are wreath products. I have procedures to get
representatives of the conjugacy classes of transitive subgroups for these
groups, that I can provide to you if you want. However, it is not hard to
classify their maximal subgroups.
Of the remaining 26 groups, 18 are of size smaller 10000. You can use the
subgroup lattice program to get their maximal subgroups.
Most of the other groups are normal of small index in a wreath product.
Using this information one can describe the transitive subgroups of them.
(Again, I have functions to deal with this case. Write to me if you want
further information.)
Remaining is M12, whose maximal subgroups are given in the ATLAS.
In all these cases, after getting the maximal subgroups, treatment is the
same as in the solvable case.

Going through this process is a tour de force. However I can't imagine an
easier way (except persuading someone else to do it, but that's exactly
what I'm doing here).

I hope this helps. If anything in my description is unclear please ask.

Alexander Hulpke

-- Lehrstuhl D fuer Mathematik, RWTH, Templergraben 64, 52056 Aachen, Germany,
eMail: Alexander.Hulpke@math.rwth-aachen.de

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