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action.jl
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action.jl
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@testset "natural stabilizers in permutation groups" begin
G = symmetric_group(5)
S = stabilizer(G, 1)
@test order(S[1]) == 24
@test S[1] == stabilizer(G, 1, ^)[1]
pt = [1, 2]
S = stabilizer(G, pt)
@test order(S[1]) == 6
@test S[1] == stabilizer(G, pt, on_tuples)[1]
S = stabilizer(G, Set(pt))
@test order(S[1]) == 12
@test S[1] == stabilizer(G, Set(pt), on_sets)[1]
l = [1, 1, 2, 2, 3]
S = stabilizer(G, l, permuted)
@test order(S[1]) == 4
# a more complex example
G = symmetric_group(14)
gens = [ cperm([1,10], [2,12,13,7,8,14], [3,4,9,6,5,11]),
cperm([1,11,6,13,12,10,2,8,4,3]) ]
H = sub(G, gens)[1]
# stabilizer should work
K = stabilizer(H, 1)[1]
@test order(K) == 46080
# bugfix test: stabilizer should still work after group size is known
@test order(H) == 645120
@test K == stabilizer(H, 1)[1]
end
@testset "natural stabilizers in matrix groups" begin
n = 3
F = GF(2)
G = general_linear_group(n, F)
V = AbstractAlgebra.Generic.FreeModule(F, n)
v = gen(V, 1)
S = stabilizer(G, v)
@test order(S[1]) == 24
@test S[1] == stabilizer(G, v, *)[1]
S = stabilizer(G, gens(V))
@test order(S[1]) == 1
@test S[1] == stabilizer(G, gens(V), on_tuples)[1]
S = stabilizer(G, Set(gens(V)))
@test order(S[1]) == 6
@test S[1] == stabilizer(G, Set(gens(V)), on_sets)[1]
end
@testset "action on multivariate polynomials: permutations" begin
g = symmetric_group(3)
R, vars = polynomial_ring(QQ, 3);
(x1, x2, x3) = vars
f = x1*x2 + x2*x3
iso = Oscar.iso_oscar_gap(R)
img = iso(f)
for p in [g(cperm(1:3)), g(cperm(1:2))]
@test f^p == evaluate(f, permuted(vars, p^-1))
@test on_indeterminates(img, p) == iso(f^p)
end
for x in gens(g), y in gens(g)
@test on_indeterminates(on_indeterminates(f, x), y) == on_indeterminates(f, x*y)
end
I = ideal(R, [x1^2*x2, x2^2])
orb = orbit(g, on_indeterminates, I)
@test length(orb) == 6
II = intersect(collect(orb)...)
@test length(orbit(g, on_indeterminates, II)) == 1
@test I^gen(g, 1) == on_indeterminates(I, gen(g, 1))
end
@testset "action on elements of free assoc. algebras: permutations" begin
g = symmetric_group(3)
R, vars = free_associative_algebra(QQ, 3);
(x1, x2, x3) = vars
f = x1*x2 + x2*x3
for p in [g(cperm(1:3)), g(cperm(1:2))]
@test f^p == evaluate(f, permuted(vars, p^-1))
end
for x in gens(g), y in gens(g)
@test on_indeterminates(on_indeterminates(f, x), y) == on_indeterminates(f, x*y)
end
end
@testset "action on multivariate polynomials: matrices" begin
g = general_linear_group(3, 5)
R, vars = polynomial_ring(base_ring(g), degree(g))
(x1, x2, x3) = vars
f = x1*x2 + x2*x3
# permutation matrix
p = cperm(1:3)
m = g(permutation_matrix(base_ring(g), p))
@test f^p == x1*x3 + x2*x3
@test f^m == f^p
iso = Oscar.iso_oscar_gap(R)
img = iso(f)
@test on_indeterminates(img, m) == iso(f^m)
# non-permutation matrix
m = g(matrix(base_ring(g), 3, 3, [3, 0, 2, 4, 0, 0, 0, 4, 0]))
@test f^m == 2*x1^2 + x1*x2 + 3*x1*x3
for x in gens(g), y in gens(g)
@test on_indeterminates(on_indeterminates(f, x), y) == on_indeterminates(f, x*y)
end
I = ideal(R, [x1^2*x2, x2^2])
orb = orbit(g, on_indeterminates, I)
@test length(orb) == 186
II = intersect(collect(orb)...)
@test length(orbit(g, on_indeterminates, II)) == 1
@test I^gen(g, 1) == on_indeterminates(I, gen(g, 1))
end
@testset "projective action on lines" begin
n = 3
F = GF(5)
G = general_linear_group(n, F)
V = AbstractAlgebra.Generic.FreeModule(F, n)
v = gen(V, 1)
v = on_lines(v, one(G)) # make sure that `v` is normalized
@test on_lines(2*v, one(G)) == v
orb = orbit(G, on_lines, v)
@test length(orb) == 31
epi = action_homomorphism(orb)
@test (order(F) - 1) * order(image(epi)[1]) == order(G)
@test_throws AssertionError on_lines(zero(V), one(G))
end