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#############################################################################
##
#W moufang_triality.gi Triality of Moufang loops [loops]
##
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
##
#F TrialityPermGroup( L )
##
## Returns the triality group associated with Moufang loop <L>,
## as a permutation group.
InstallGlobalFunction( TrialityPermGroup, function( L )
local AA, BB, L_inv, i, trg_perm, trrho, trsigma;
if IsGroup( L ) then L:=IntoLoop( L ); fi;
if not( IsMoufangLoop( L ) ) then
Error( "LOOPS: <1> has to be a Moufang loop." );
fi;
trrho := Product( List( [1..Size(L)], k -> (k,k+Size(L),k+2*Size(L)) ) );
L_inv := PermList( List( [1..Size(L)], k -> 1^(LeftSection(L)[k]^-1) ) );
trsigma := L_inv^(trrho^2) *
Product( List( [1..Size(L)], k ->
(k, Size(L)+k^L_inv ) ) );
AA := [];
BB := [];
for i in PosInParent( GeneratorsSmallest( L ) ) do
Add( AA, (LeftSection(L)[i]*RightSection(L)[i])^-1
* LeftSection(L)[i]^trrho
* RightSection(L)[i]^(trrho^2) );
od;
BB := OnTuples( AA, trrho );
trg_perm := Group( Concatenation( AA, BB ) );
SetName(trg_perm, Concatenation(
"<Triality (dual collineation) group of order ",
StringPP( Order( trg_perm ) ),
">"
));
return rec( group := trg_perm, sigma := trsigma, rho := trrho );
end);
#############################################################################
##
#F TrialityPcGroup( L )
##
## Returns the triality group associated with Moufang loop <L>,
## as a pc group.
InstallGlobalFunction( TrialityPcGroup, function( L )
local trcoll, trcoll_pcgs, pc_gens, big_gr, big_pcgs,
trg_pc,sigma_pc,rho_pc;
if not( IsSolvable( MultiplicationGroup( L ) ) ) then
Error( "LOOPS: The triality group is not solvable." );
fi;
trcoll := TrialityPermGroup( L );
if IsNilpotent( L ) then
trcoll_pcgs := SpecialPcgs( trcoll.group );
else
trcoll_pcgs := Pcgs( trcoll.group );
fi;
pc_gens := PcgsByPcSequence(
FamilyObj( One( trcoll.group ) ),
Concatenation(
[ trcoll.sigma, trcoll.rho ],
trcoll_pcgs
)
);
big_gr := PcGroupWithPcgs( pc_gens );
big_pcgs := Pcgs( big_gr );
trg_pc := Group( big_pcgs{[3..Length(big_pcgs)]} );
sigma_pc := big_pcgs[1];
rho_pc := big_pcgs[2];
SetName(trg_pc, Concatenation(
"<Triality pc group of order ",
StringPP( Order( trg_pc ) ),
">"
));
return rec( group := trg_pc, sigma := sigma_pc, rho := rho_pc );
end);
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