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gsets.jl
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gsets.jl
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@testset "G-sets of permutation groups" begin
# natural constructions (determined by the types of the seeds)
G = symmetric_group(6)
Omega = gset(G)
@test repr(Omega, context = :supercompact => true) == "G-set"
@test isa(Omega, GSet)
@test length(Omega) == 6
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
@test collect(Omega) == 1:6 # ordering is kept
Omega = gset(G, [Set([1, 2])]) # action on unordered pairs
@test isa(Omega, GSet)
@test length(Omega) == 15
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
Omega = gset(G, [[1, 2]]) # action on ordered pairs
@test isa(Omega, GSet)
@test length(Omega) == 30
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
Omega = gset(G, [(1, 2)]) # action on ordered pairs (repres. by tuples)
@test isa(Omega, GSet)
@test length(Omega) == 30
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
# constructions by explicit action functions
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
@test isa(Omega, GSet)
@test length(Omega) == 740
@test length(orbits(Omega)) == 2
@test ! is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
R, x = polynomial_ring(QQ, ["x1", "x2", "x3"]);
f = x[1]*x[2] + x[2]*x[3]
G = symmetric_group(3)
Omega = gset(G, on_indeterminates, [f])
@test isa(Omega, GSet)
@test length(Omega) == 3
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
# seeds can be anything iterable
G = symmetric_group(6)
@test isa(gset(G, 1:6), GSet)
@test isa(gset(G, collect(1:6)), GSet)
@test isa(gset(G, Set(1:6)), GSet)
# basic functionality
G = symmetric_group(6)
Omega = gset(G, [Set([1, 2])])
@test representative(Omega) in Omega
@test acting_domain(Omega) == G
# wrapped elements of G-sets
G = symmetric_group(4)
omega = [1, 2]
Omega = gset(G, Set([omega])) # action on ordered pairs
g = gen(G, 1)
x = Omega(omega)
@test x in Omega
@test unwrap(x) == omega
@test unwrap(omega) == omega
@test x^g == Omega(omega^g) # action via `^` is defined for both
@test length(orbit(x)) == 12
omega = [0,1,0,1]
Omega = gset(G, permuted, Set([omega]))
g = gen(G, 1)
x = Omega(omega)
@test x in Omega
@test unwrap(x) == omega
@test unwrap(omega) == omega
@test x^g == Omega(permuted(omega, g)) # action via `^` is defined for `x`
@test_throws ErrorException omega^g # ... but not for the unwrapped object
orb = orbit(x)
@test length(orb) == 6
@test isa(orb, GSet)
# construction from a known set
G = sylow_subgroup(symmetric_group(4), 3)[1]
given = 1:4
Omega = as_gset(G, given)
@test length(Omega) == 4
@test length(orbits(Omega)) == 2
@test isa(Omega, GSet)
@test isa(orbit(Omega, 1), GSet)
@test collect(Omega) == given # ordering is kept
# orbit
G = symmetric_group(6)
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
@test length(orbit(Omega, [0,1,0,1,0,1])) == length(Oscar.orbit_via_Julia(Omega, [0,1,0,1,0,1]))
# permutation
G = symmetric_group(6)
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
g = gen(G, 1)
pi = permutation(Omega, g)
@test order(pi) == order(g)
@test degree(parent(pi)) == length(Omega)
# action homomorphism
G = symmetric_group(6)
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
acthom = action_homomorphism(Omega)
@test pi == g^acthom
@test has_preimage_with_preimage(acthom, pi)[1]
@test order(image(acthom)[1]) == 720
rest = restrict_homomorphism(acthom, derived_subgroup(G)[1])
@test ! is_bijective(rest)
# is_conjugate
G = symmetric_group(6)
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
@test is_conjugate(Omega, [0,1,0,1,0,1], [1,0,1,0,1,0])
@test ! is_conjugate(Omega, [0,1,0,1,0,1], [1,2,3,4,5,6])
# is_conjugate_with_data
G = symmetric_group(6)
Omega = gset(G, permuted, [[0,1,0,1,0,1], [1,2,3,4,5,6]])
rep = is_conjugate_with_data(Omega, [0,1,0,1,0,1], [1,0,1,0,1,0])
@test rep[1]
@test permuted([0,1,0,1,0,1], rep[2]) == [1,0,1,0,1,0]
rep = is_conjugate_with_data(Omega, [0,1,0,1,0,1], [1,2,3,4,5,6])
@test ! rep[1]
end
@testset "natural action of permutation groups" begin
G8 = transitive_group(8, 3)
S4 = symmetric_group(4)
@test order(G8) == 8
# all_blocks
bl = all_blocks(G8)
@test length(bl) == 14
@test [1, 2] in bl
@test length(all_blocks(S4)) == 0
# blocks
bl = blocks(G8)
@test elements(bl) == [[1, 8], [2, 3], [4, 5], [6, 7]]
@test elements(bl) == elements(blocks(G8, 1:degree(G8)))
# block homomorphism
blhom = action_homomorphism(bl)
@test ! is_bijective(blhom)
# is_primitive
@test ! is_primitive(G8)
@test ! is_primitive(G8, 1:degree(G8))
@test is_primitive(S4)
@test ! is_primitive(S4, 1:3)
# is_regular
@test is_regular(G8)
@test ! is_regular(G8, 1:9)
@test ! is_regular(S4)
# is_semiregular
@test is_semiregular(G8)
@test ! is_semiregular(G8, 1:9)
@test ! is_semiregular(S4)
# is_transitive
@test is_transitive(G8)
@test ! is_transitive(G8, 1:9)
# maximal_blocks
bl = maximal_blocks(G8)
@test elements(bl) == [[1, 2, 3, 8], [4, 5, 6, 7]]
@test elements(bl) == elements(maximal_blocks(G8, 1:degree(G8)))
@test elements(maximal_blocks(S4)) == [[1, 2, 3, 4]]
# minimal_block_reps
bl = minimal_block_reps(G8)
@test bl == [[1,i] for i in 2:8]
@test bl == minimal_block_reps(G8, 1:degree(G8))
@test minimal_block_reps(S4) == [[1, 2, 3, 4]]
# transitivity
@test transitivity(G8) == 1
@test transitivity(S4) == 4
@test_throws ArgumentError transitivity(S4, 1:3)
@test transitivity(S4, 1:4) == 4
@test transitivity(S4, 1:5) == 0
end
@testset "G-sets of matrix groups over finite fields" begin
# natural constructions (determined by the types of the seeds)
G = general_linear_group(2, 3)
V = free_module(base_ring(G), degree(G))
Omega = gset(G)
@test isa(Omega, GSet)
@test length(Omega) == 9
@test length(orbits(Omega)) == 2
@test ! is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
@test collect(Omega) == collect(V) # ordering is kept
Omega = orbit(G, gen(V, 1))
@test isa(Omega, GSet)
@test length(Omega) == 8
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
Omega = gset(G, [Set(gens(V))]) # action on unordered pairs of vectors
@test isa(Omega, GSet)
@test length(Omega) == 24
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
Omega = gset(G, [gens(V)]) # action on ordered pairs of vectors
@test isa(Omega, GSet)
@test length(Omega) == 48
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test is_regular(Omega)
@test is_semiregular(Omega)
# orbit
Omega = gset(G)
v = gen(V, 1)
@test length(orbit(Omega, v)) == length(Oscar.orbit_via_Julia(Omega, v))
# permutation
Omega = gset(G)
g = gen(G, 1)
pi = permutation(Omega, g)
@test order(pi) == order(g)
@test degree(parent(pi)) == length(Omega)
# action homomorphism
Omega = gset(G)
acthom = action_homomorphism(Omega)
@test pi == g^acthom
@test has_preimage_with_preimage(acthom, pi)[1]
@test order(image(acthom)[1]) == 48
# is_conjugate
Omega = gset(G)
@test is_conjugate(Omega, gen(V, 1), gen(V, 2))
@test ! is_conjugate(Omega, zero(V), gen(V, 1))
# is_conjugate_with_data
Omega = gset(G)
rep = is_conjugate_with_data(Omega, gens(V)...)
@test rep[1]
@test gen(V, 1) * rep[2] == gen(V, 2)
rep = is_conjugate_with_data(Omega, zero(V), gen(V, 1))
@test ! rep[1]
end
@testset "orbits of matrix groups over finite fields" begin
@testset for F in [ GF(2), GF(3), GF(2,2) ], n in 2:4
q = order(F)
V = vector_space(F, n)
GL = general_linear_group(n, F)
S = sylow_subgroup(GL, 2)[1]
for G in [GL, S]
# for k in 0:n # k = 0 is a problem in GAP 4.12.0
for k in 1:n
res = orbit_representatives_and_stabilizers(G, k)
total = ZZ(0)
for (U, stab) in res
total = total + index(G, stab)
@test length(orbit(stab, U)) == 1
end
num = ZZ(1)
for i in 0:(k-1)
num = num * (q^n - q^i)
end
for i in 0:(k-1)
num = divexact(num, q^k - q^i)
end
@test total == num
end
end
end
end
@testset "G-sets of matrix groups in characteristic zero" begin
# natural constructions (determined by the types of the seeds)
G = matrix_group(permutation_matrix(QQ,[3,1,2]))
R, (x,y,z) = polynomial_ring(QQ, [:x,:y,:z])
f = x^2 + y
orb = orbit(G, f)
@test length(orb) == 3
end
@testset "G-sets by right transversals" begin
G = symmetric_group(5)
H = sylow_subgroup(G, 2)[1]
Omega = right_cosets(G, H)
@test repr(Omega, context = :supercompact => true) == "Right cosets of groups"
@test isa(Omega, GSet)
@test acting_group(Omega) == G
@test length(Omega) == index(G, H)
@test Omega[end] == Omega[length(Omega)]
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
@test eltype(Omega) == typeof(representative(Omega))
# iteration
for i in 1:length(Omega)
@test findfirst(is_equal(Omega[i]), Omega) == i
end
rep = representative(Omega)
for omega in Omega
@test one(G) in omega || omega != rep
end
# orbit
g = gen(G, 1)
pnt = right_coset(H, g)
@test pnt in Omega
@test length(orbit(Omega, pnt)) == length(Oscar.orbit_via_Julia(Omega, pnt))
# permutation
pi = permutation(Omega, g)
@test order(pi) == order(g)
@test degree(parent(pi)) == length(Omega)
fun = Oscar.action_function(Omega)
for i in 1:length(Omega)
@test Omega[i^pi] == fun(Omega[i], g)
end
# action homomorphism
acthom = action_homomorphism(Omega)
@test pi == g^acthom
flag, pre = has_preimage_with_preimage(acthom, pi)
@test flag
@test pre == g
@test order(image(acthom)[1]) == order(G)
rest = restrict_homomorphism(acthom, derived_subgroup(G)[1])
@test ! is_bijective(rest)
# is_conjugate
x, y = [right_coset(H, g) for g in gens(G)]
@test is_conjugate(Omega, x, y)
# is_conjugate_with_data
rep = is_conjugate_with_data(Omega, x, y)
@test rep[1]
@test x * rep[2] == y
@test Oscar.action_function(Omega)(x, rep[2]) == y
end
@testset "G-sets by left transversals" begin
G = symmetric_group(5)
H = sylow_subgroup(G, 2)[1]
Omega = left_cosets(G, H)
@test repr(Omega, context = :supercompact => true) == "Left cosets of groups"
@test isa(Omega, GSet)
@test acting_group(Omega) == G
@test length(Omega) == index(G, H)
@test Omega[end] == Omega[length(Omega)]
@test length(orbits(Omega)) == 1
@test is_transitive(Omega)
@test ! is_regular(Omega)
@test ! is_semiregular(Omega)
@test eltype(Omega) == typeof(representative(Omega))
# iteration
for i in 1:length(Omega)
@test findfirst(is_equal(Omega[i]), Omega) == i
end
rep = representative(Omega)
for omega in Omega
@test one(G) in omega || omega != rep
end
# orbit
g = gen(G, 1)
pnt = left_coset(H, g)
@test pnt in Omega
@test length(orbit(Omega, pnt)) == length(Oscar.orbit_via_Julia(Omega, pnt))
# permutation
pi = permutation(Omega, g)
@test order(pi) == order(g)
@test degree(parent(pi)) == length(Omega)
fun = Oscar.action_function(Omega)
for i in 1:length(Omega)
@test Omega[i^pi] == fun(Omega[i], g)
end
# action homomorphism
acthom = action_homomorphism(Omega)
@test pi == g^acthom
flag, pre = has_preimage_with_preimage(acthom, pi)
@test flag
@test pre == g
@test order(image(acthom)[1]) == order(G)
rest = restrict_homomorphism(acthom, derived_subgroup(G)[1])
@test ! is_bijective(rest)
# is_conjugate
x, y = [left_coset(H, g) for g in gens(G)]
@test is_conjugate(Omega, x, y)
# is_conjugate_with_data
rep = is_conjugate_with_data(Omega, x, y)
@test rep[1]
@test inv(rep[2]) * x == y
@test Oscar.action_function(Omega)(x, rep[2]) == y
end