[GAP Forum] Obtaining Small Group information

[Heiko Dietrich]

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
> >      >
> >      > _______________________________________________
> >      > Forum mailing list
> >      > Forum at mail.gap-system.org <mailto:Forum at mail.gap-system.org>
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> >
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-- 

Dipl. Math. Heiko Dietrich                   
Department of Mathematics
University Braunschweig
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GERMANY

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