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#############################################################################
##
#W psoluble.gi Permutability GAP library ABB&ECL&RER
##
##
#Y Copyright (C) 2000-2018 Adolfo Ballester-Bolinches, Enric Cosme-Ll\'opez
#Y and Ramon Esteban-Romero
##
## This file contains methods for p-soluble, p-supersoluble,
## and p-nilpotent groups
##
#############################################################################
##
#M IsPSupersolvable( <G>, <p> ) . . . . for finite groups
##
## Returns true if the group is p-supersolvable, false otherwise.
## A group $G$ is p-supersolvable if all chief factors of
## order divisible by p are cyclic.
##
##
InstallMethod(IsPSupersolvableOp,
"for finite supersolvable groups",
[IsGroup and IsFinite and IsSupersolvableGroup,IsPosInt],
function(G,p)
return true;
end);
InstallMethod(IsPSupersolvableOp,
"method for a finite group with chief series and prime",
[IsGroup and IsFinite and HasChiefSeries,IsPosInt],function(G,p)
local cs,x,fact,j,l;
if Size(G) mod p <>0
then
return true;
fi;
cs:=ChiefSeries(G);
l:=List([1..Length(cs)-1],j->Index(cs[j],cs[j+1]));
return ForAll([1..Length(cs)-1],
j->(l[j]=p) or
(l[j] mod p <> 0));
end);
InstallMethod(IsPSupersolvableOp,
"generic method for a finite group and prime",
[IsGroup and IsFinite,IsPosInt],function(G,p)
local cs,x,fact;
if (Size(G) mod p) <> 0 or IsSupersolvable(G)
then
return true;
fi;
cs:=ChiefSeries(G);
return IsPSupersolvable(G,p);
end);
#############################################################################
##
#M IsSylowTowerGroup( <G> ) . . . . for finite groups
##
## Returns true if the group has a Sylow tower of supersolvable type,
## else returns false
##
InstallMethod(IsSylowTowerGroup,"for finite groups",
[IsGroup and IsFinite],
function(g)
local n,pr,l;
pr:=PrimesDividingSize(g);
l:=Length(pr);
return IsSolvableGroup(g) and
ForAll([1..l],n->IsNormal(g, HallSubgroup(g, pr{[n..l]})));
end);
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