[GAP Forum] Obtaining Small Group information
Heiko Dietrich
H.Dietrich at tu-bs.de
Fri Dec 12 06:42:22 GMT 2008
Hello,
the groups of order 5^3 are the following
(1) cyclic group C_{125}
(2) abelian group C_{25} x C_5
(3) extraspecial 5-group of order 125 and exponent 5: C_5 \ltimes (C_5 x C_5)
(4) extraspecial 5-group of order 125 and exponent 25: C_5 \ltimes C_{25}
(5) elementary abelian group C_5 x C_5 x C_5.
(Note that A \ltimes B is a split extensions with A acting on the normal
subgroup B.)
In general, the groups of order p^3 were classified by Otto Hoelder, "Die
Gruppen der Ordnungen p^3 , pq^2, pqr, p^4.", Math. Ann., 43: 301 - 412,
1893.
As 1625 = 13*5^3, a group G of order 1625 has a normal Sylow subgroup P of
order 125. Now Schur-Zassenhaus show that P has a complement in G, that is, G
is isomorphic to the split extension C_{13} \ltimes P. Due to order reasons,
the cyclic group has to act trivially, that is, G is isomorphic to the direct
product C_{13} x P.
Hence, you obtain all groups of order 1625 by adding a direct factor C_{13} to
every group (1)--(5) of order 125:
gap>NumberSmallGroups(1625);
5
gap>List(AllSmallGroups(125),x->IdSmallGroup(DirectProduct(x,CyclicGroup(13))));
[ [ 1625, 1 ], [ 1625, 2 ], [ 1625, 3 ], [ 1625, 4 ], [ 1625, 5 ] ]
Hope this helps,
Heiko
On Thursday 11 December 2008 23:19, Paweł Laskoś-Grabowski wrote:
> Hello,
>
> Much of this (highly useful otherwise, thanks a lot) information is
> actually much more general than I need at the moment. I need to know the
> structure of all (up to isomorphism, of course) groups of orders 125 and
> 1625. I was glad to discover that there are only five of each, but now
> it seems that the ones obtained by semidirect products may actually
> represent many non-isomorphic groups. Is there a way to obtain such
> level of details using GAP, or should I refer to textbooks and/or prove
> few facts myself to get the information I need?
>
> Regards,
> Pawel Laskos-Grabowski
>
> Joe Bohanon schrieb:
> > I would point out that StructureDescription might not always return a
> > group the way you'd like it. The manual explains a little more about
> > how it picks a particular form for the structure.
> >
> > That function also does not do anything with central products. Hence if
> > I type:
> > StructureDescription(SmallGroup(32,50)) I get:
> > "(C2 x Q8) : C2" when it's also a central product of Q8 with D8. It
> > returns some pretty awkward answers for other larger central products.
> >
> > It also will usually not let you know how the split or non-split
> > extensions work, so you might get two non-isomorphic groups that return
> > the same "StructureDescription".
> >
> > Also be forewarned that many times GAP will just compute the whole
> > subgroup lattice to find a structure, so any group that would take a
> > long time with LatticeByCyclicExtension or ConjugacyClassesSubgroups is
> > likely to take a long time for StructureDescription. This would
> > include, for instance, 2-groups of rank more than 5, groups with large
> > permutation representations or large matrix representations and also
> > finitely-presented groups. It does have a separate routine for any
> > simple group that spits out the answer due to the classification in
> > almost no time, however, while it could easily tell me a group is
> > isomorphic to, say U4(3), it would take much longer (and probably use up
> > all of your RAM) to say a group is isomorphic to U4(3):D8.
> >
> > On Thu, Dec 11, 2008 at 6:37 AM, Heiko Dietrich <h.dietrich at tu-bs.de
> > <mailto:h.dietrich at tu-bs.de>> wrote:
> >
> > Dear Paweł,
> >
> > you can use the command "StructureDescription":
> >
> > gap> for i in AllSmallGroups(1625) do
> > Display(StructureDescription(i)); od;
> > C1625
> > C325 x C5
> > C13 x ((C5 x C5) : C5)
> > C13 x (C25 : C5)
> > C65 x C5 x C5
> >
> > The output is explained in the manual:
> >
> > http://www.gap-system.org/Manuals/doc/htm/ref/CHAP037.htm#SECT006
> >
> > Best,
> > Heiko
> >
> > On Tuesday 09 December 2008 20:56, Paweł Laskoś-Grabowski wrote:
> > > Hello,
> > >
> > > I have noticed that GAP Small Groups library provides useful
> >
> > information
> >
> > > on the structure of groups belonging to the layer 1 of the
> >
> > library, but
> >
> > > does not do so for (some) bit more complicated groups. I am rather
> > > dissatisfied by the output
> > >
> > > gap> SmallGroupsInformation(1625);
> > >
> > > There are 5 groups of order 1625.
> > > They are sorted by normal Sylow subgroups.
> > > 1 - 5 are the nilpotent groups.
> > >
> > > How can I obtain such a pleasant info like the following?
> > >
> > > gap> SmallGroupsInformation(125);
> > >
> > > There are 5 groups of order 125.
> > > 1 is of type c125.
> > > 2 is of type 5x25.
> > > 3 is of type 5^2:5.
> > > 4 is of type 25:5.
> > > 5 is of type 5^3.
> > >
> > > And, by the way, what does the colon stand for in the 125,3 and
> > > 125,4 type descriptions? I failed to find the explanation in the
> > > help
> >
> > pages.
> >
> > > Regards,
> > > Paweł Laskoś-Grabowski
> > >
> > > _______________________________________________
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> >
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--
Dipl. Math. Heiko Dietrich
Department of Mathematics
University Braunschweig
Pockelsstrasse 14
38106 Braunschweig
GERMANY
Room: F 613
Tel: ++49 (0)531 391 7526
Fax: ++49 (0)531 391 8206
Email: h.dietrich at tu-bs.de
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