[GAP Forum] Generalized Fitting Subgroup

[Stefanos Aivazidis]

Dear Alex & forum,

I want to offer a naive solution to your question. I hope
that more experienced users will "chip in" to improve it.

Best wishes,
Stefanos


#Check whether a group is quasi-simple, i.e., perfect whose quotient by its
centre is (non-abelian) simple

IsQuasisimpleGroup:=function(g)

local quo;

if IsSolvableGroup(g) then

return false;

fi;

if IsPerfectGroup(g) then

quo:=FactorGroup(g,Center(g));

if IsSimpleGroup(quo) then

return true;

fi;

fi;

return false;

end;;


#Construct the layer of a group, that is, the product of its subnormal
quasi-simple subgroups

Layer:=function(g)

local list,h,lay;

if IsSolvableGroup(g) then

return TrivialSubgroup(g);

fi;

if IsSimpleGroup(g) then

return g;

fi;

list:=[];;

for h in AllSubgroups(g) do

if IsSubnormal(g,h) then

if IsQuasisimpleGroup(h) then

Append(list,[h]);

fi;

fi;

od;

lay:=TrivialSubgroup(g);

for h in list do

lay:=ClosureGroup(lay,h);

od;

return lay;

end;;


#Finally, construct the generalised Fitting subgroup of the group

GeneralizedFittingSubgroup:=function(g)

local fitt, lay;

fitt:=FittingSubgroup(g);

lay:=Layer(g);

return ClosureGroup(fitt,lay);

end;;



On 6 January 2018 at 22:02, Alex Trofimuk <trof1984 at rambler.ru> wrote:

> Dear Forum,
> Alex Trofimuk asked:
> Whether exists in GAP a function for finding a generalized Fitting
> subgroup for
> an arbitrary group?
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