[GAP Forum] Small Groups which are not a direct product of a smaller group and a cyclic group

[Horvath Gabor]

Hi Jacek,

Direct product decomposition is unique up to isomorphism of the factors. 
So if StructureDescription says Csomething x smallgerGroup, then it is 
indeed a direct product of such. When looking at decomposing further (into 
semidirect products and extensions), then the decompositon may not be 
unique, anymore. But for direct products it is unique.

The bound of 100 in the manual speaks about how quickly can one get the 
results. With GAP 4.9.3 the algorithm for StructureDescription is smarter 
than before, and with today's computers it may take faster to compute 
the structure for some groups than previously, but it still can be very 
slow depending on the group.

If you are only interested in the direct factors, then you can use the 
function DirectFactorsOfGroup, which gives you the direct factors of the 
particular group. Then you can check if any of them cyclic.

Alternatively, if G = H x <g>, then g must commute with H, and of course 
with g, thus g commutes with G, as well. Therefore, g is in the center of 
G. Furthermore, <g> must have trivial intersection with the derived 
subgroup G'. 
So you can go over the elements (or rational classes) of the center, see 
if the generated group intersects trivially the derived subgroup, and if 
yes, then you can try to find a complement (using the command 
NormalComplement). This is how DirectFactorsOfGroup finds the abelian 
direct factors. You can use something like this:


C := Center(G);
Gd := DerivedSubgroup(G);
D := Intersection(C, Gd);
for g in RationalClasses(C) do
   N := Subgroup(C, [Representative(g)]);
   if not IsTrivial(N) and IsTrivialNormalIntersection(C, D, N) then
     B := NormalComplement(G, N);
     if B <> fail then
       print(N, B);
       break;
     fi;
   fi;
od;


Hope this helps.

Best,
Gabor

On Sun, 23 Sep 2018, Jacek M. Holeczek wrote:

> Hi,
> I am looking for a bullet-proof method of identifying Small Groups that 
> cannot be written as a direct product of a smaller group and a cyclic group 
> of some size (I do not care about the size of this cyclic group).
>
> Some people use the output of the "StructureDescription(G)" method (where 
> "G:=SmallGroup(o,i);", of course) and, as soon as they see "C<size> x 
> SomeThing", they assume that the group G is such a "genuine" direct product 
> (denoted by the "x") of a "genuine" cyclic group "C<size>" and some smaller 
> group described by "SomeThing".
>
> However, the manual explicitly says that the output of this method should be 
> used for "educational" purposes only as it really provides a partial 
> description of the structure only, especially for orders higher than 100.
> I think I have even read somewhere that the "C<size>" does not necessarily 
> represent a "genuine" cyclic group and that the "x" does not necessarily 
> represent a "genuine" direct product operation.
>
> Thanks in advance,
> Best regards,
> Jacek.
>
> _______________________________________________
> Forum mailing list
> Forum at gap-system.org
> https://mail.gap-system.org/mailman/listinfo/forum
>


 						Horvath Gabor
-------------------------------------------------------------------------------
e-mail:	ghorvath at science.unideb.hu
phone: +36 52 512900 / 22798
web: http://www.math.unideb.hu/horvath-gabor