|
#############################################################################
##
#W tests.gi DifSets Package Dylan Peifer
##
## Functions check if a set/sum is a difference set/sum and determine if two
## sets/sums are equivalent.
##
#############################################################################
##
#F IsDifferenceSet( <G>, <D> )
##
InstallGlobalFunction( IsDifferenceSet, function (G, D)
local v, k, lambda, e, count, i, j, index;
v := Size(G);
k := Length(D);
lambda := k*(k-1)/(v-1);
e := Elements(G);
count := List([1..v], x->0); # times each group element is a difference
for i in D do
for j in D do
# compute the element given by this difference and add to count
index := Position(e, e[i]*e[j]^(-1));
count[index] := count[index] + 1;
# if a nonidentity element exceeds lambda then D cannot be a difset
if not index = 1 and count[index] > lambda then
return false;
fi;
od;
od;
# if counts of nonidentities never exceed lambda they must be lambda
return true;
end );
#############################################################################
##
#F IsDifferenceSum( <G>, <N>, <S> )
##
InstallGlobalFunction( IsDifferenceSum, function (G, N, S)
local v, k, lambda, w, e, count, i, j, index;
v := Size(G);
k := Sum(S);
lambda := k*(k-1)/(v-1);
w := Size(N);
# first test based on differences giving the identity
if not Sum(S,x->x^2) = lambda*w + k - lambda then
return false;
fi;
# then test all other elements
e := Elements(G/N);
count := List([1..v/w], x->0);
for i in [1..v/w] do
for j in [1..v/w] do
# compute the element given by this difference and add coeff to count
index := Position(e, e[i]*e[j]^(-1));
count[index] := count[index] + S[i]*S[j];
# if a nonidentity element exceeds lambda*w then S cannot be a difsum
if not index = 1 and count[index] > lambda*w then
return false;
fi;
od;
od;
# if counts of nonidentities never exceed lambda*w they must be lambda*w
return true;
end );
#############################################################################
##
#F IsEquivalentDifferenceSet( <G>, <D1>, <D2> )
##
InstallGlobalFunction( IsEquivalentDifferenceSet, function (G, D1, D2)
local v, prod, phi, aut, i, D;
v := Size(G);
prod := ProductTable(G);
# iterate over all automorphisms
for phi in AutomorphismGroup(G) do
aut := AutomorphismTable(G, phi);
# iterate over all group elements
for i in [1..v] do
# map D2 to g * phi(D2) and check if equal to D1
D := List(D2, x->prod[i][aut[x]]);
if IsEqualSet(D1, D) then
return true;
fi;
od;
od;
# no equivalence was found
return false;
end );
#############################################################################
##
#F IsEquivalentDifferenceSum( <G>, <N>, <S1>, <S2> )
##
InstallGlobalFunction( IsEquivalentDifferenceSum, function (G, N, S1, S2)
local v, w, theta, H, prod, phi, aut, i, S, j;
v := Size(G);
w := Size(N);
theta := NaturalHomomorphismByNormalSubgroup(G, N);
H := Image(theta);
prod := ProductTable(H);
# iterate over all induced automorphisms
for phi in AutomorphismGroup(G) do
if Image(phi, N) = N then
aut := AutomorphismTable(H, InducedAutomorphism(theta, phi));
# iterate over all group elements
for i in [1..v/w] do
# map S2 to g * aut(S2) and check if equal to S1
S := List([1..v/w]);
for j in [1..v/w] do
S[prod[i][aut[j]]] := S2[j];
od;
if S1 = S then
return true;
fi;
od;
fi;
od;
# no equivalence was found
return false;
end );
#############################################################################
##
#E
[ Dauer der Verarbeitung: 0.28 Sekunden
(vorverarbeitet)
]
|
