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#############################################################################
##
#W bol_core_methods.gi Common methods for Bol loops [loops]
##
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
##
#A AssociatedLeftBruckLoop( Q )
##
## Given a left Bol loop Q for which x -> x^2 is a permutation,
## returns the associated left Bruck loop, defined by
## x o y = (x*((y*y)*x))^(1/2) (default) or
## x o y = x^(1/2)*(y*x^(1/2)) (commented out)
InstallMethod( AssociatedLeftBruckLoop,
[ IsLeftBolLoop ],
function( Q )
local n, ct, squares, roots, new_ct, L, i, j;
n := Size( Q );
ct := CanonicalCayleyTable( CayleyTable( Q ) );
squares := List( [ 1..n ], i -> ct[i][i] );
if not Size( Set( squares ) ) = n then
Error( "LOOPS: <1> must be a Bol loop in which squaring is a bijection." );
fi;
roots := Inverse( PermList( squares ) ); # square roots
new_ct := List([1..n], i -> [1..n] );
for i in [1..n] do for j in [1..n] do
new_ct[i][j] := (ct[i][ct[ct[j][j]][i]])^roots; # (x*((y*y)*x))^(1/2)
# new_ct[i][j] := ct[i^roots][ct[ j ][ i^roots ]]; # x^(1/2)*(y*x^(1/2))
od; od;
L := LoopByCayleyTable( new_ct ); # the associated left Bruck loop
SetIsLeftBruckLoop( L, true );
return L;
end );
#############################################################################
##
#A AssociatedRightBruckLoop( Q )
##
## Given a right Bol loop Q for which x -> x^2 is a permutation,
## returns the associated right Bruck loop, defined by
## x o y = ((x*(y*y))*x)^(1/2) (default) or
## x o y = (x^(1/2)*y)*x^(1/2) (commented out)
InstallMethod( AssociatedRightBruckLoop,
[ IsRightBolLoop ],
function( Q )
local n, ct, squares, roots, new_ct, L, i, j;
n := Size( Q );
ct := CanonicalCayleyTable( CayleyTable( Q ) );
squares := List( [ 1..n ], i -> ct[i][i] );
if not Size( Set( squares ) ) = n then
Error( "LOOPS: <1> must be a right Bol loop in which squaring is a bijection." );
fi;
roots := Inverse( PermList( squares ) ); # square roots
new_ct := List([1..n], i -> [1..n] );
for i in [1..n] do for j in [1..n] do
new_ct[i][j] := (ct[ct[j][ct[i][i]]][j])^roots; # x o y = ((y*(x*x))*y)^(1/2)
# new_ct[i][j] := ct[ct[j^roots][i]][j^roots]; # x o y = (y^(1/2)*x)*y^(1/2)
od; od;
L := LoopByCayleyTable( new_ct ); # the associated right Bruck loop
SetIsRightBruckLoop( L, true );
return L;
end );
#############################################################################
##
#O IsExactGroupFactorization( G, H1, H2 )
##
## Let G be a group and H_1, H_2 subgroups. The triple (G,H_1,H_2) is an
## exact group factorization, if H_1 \cap H_2 = 1 and G=H_1H_2.
##
## Returns true if (G, H1, H2) is an exact group factorization.
InstallMethod( IsExactGroupFactorization, "for a group and two subgroups",
[ IsGroup, IsGroup, IsGroup ],
function( G, H1, H2 )
return IsSubgroup(G,H1) and IsSubgroup(G,H2) and Size(G)=Size(H1)*Size(H2) and IsTrivial(Intersection(H1,H2));
end);
#############################################################################
##
#F RightBolLoopByExactGroupFactorization
##
## Let (G,H_1,H_2) be an exact group factorization. Define
## U = G\times G, S = H_1 \times H_2, and T = {(g,g^{-1}) : g in G }.
## Then (U,S,T) is a right Bol loop folder
## Returns the right Bol loop corresponding to (U,S,T).
InstallGlobalFunction( RightBolLoopByExactGroupFactorizationNC,
function( g, h1, h2 )
local f,sect,stab,ghom,st,q,rmlt;
f := DirectProduct( g, g );
sect := List( g, x -> Image( Embedding( f, 1 ), x )*Image( Embedding( f, 2 ), x^-1 ) );
stab := ClosureGroup( Image( Embedding( f, 1 ), h1 ), Image( Embedding( f, 2 ), h2 ) );
ghom := ActionHomomorphism( f, RightCosets( f, stab ), OnRight );
st := Image( ghom, sect );;
q := LoopByRightSection( st );
SetRightSection( q, st );
rmlt := Subgroup( f, sect );
SetRightMultiplicationGroup( q, Image( ghom, Subgroup( f, SmallGeneratingSet( rmlt ) ) ) );
return q;
end);
InstallGlobalFunction( RightBolLoopByExactGroupFactorization,
function( arg )
local g, h1, h2;
if Length( arg ) = 3 then
g := arg[1]; h1 := arg[2]; h2 := arg[3];
elif Length( arg ) = 2 then
g := arg[1]; h1 := arg[2]; h2 := Stabilizer( g, Orbit( g )[1] );
elif Length( arg ) = 1 and Length( arg[1] ) = 3 then
g := arg[1][1]; h1 := arg[1][2]; h2 := arg[1][3];
elif Length(arg) = 1 and Length( arg[1] ) = 2 then
g := arg[1][1]; h1 := arg[1][2]; h2 := Stabilizer(g,Orbit(g)[1]);
else
Error("LOOPS: Argument must be an exact group factorization or a group with a regular subgroup.");
fi;
if not IsExactGroupFactorization( g, h1, h2 ) then
Error("LOOPS: Argument does not correspond to an exact group factorization.");
fi;
return RightBolLoopByExactGroupFactorizationNC( g, h1, h2 );
end);
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