Also in general the groups of order p^3*q were classified by Western
near the beginning of the 20th century (if you have Burnside's book, you
can find the reference there where he does p^2 q). I think very few
generic classifications were done beyond what Hoelder did (except for
the work on p-groups by O'Brien and Eick) mainly because of the number
of sub-cases you have to look at.
Heiko Dietrich wrote:
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.dietr...@tu-bs.de
<mailto:h.dietr...@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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