def test_minimal_block(): D = DihedralGroup(6) block_system = D.minimal_block([0,3]) for i in range(3): assert block_system[i] == block_system[i+3] S = SymmetricGroup(6) assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0]
def test_normal_closure(): # the normal closure of the trivial group is trivial S = SymmetricGroup(3) identity = Permutation([0, 1, 2]) closure = S.normal_closure(identity) assert closure.is_trivial # the normal closure of the entire group is the entire group A = AlternatingGroup(4) assert A.normal_closure(A).is_subgroup(A) # brute-force verifications for subgroups for i in (3, 4, 5): S = SymmetricGroup(i) A = AlternatingGroup(i) D = DihedralGroup(i) C = CyclicGroup(i) for gp in (A, D, C): assert _verify_normal_closure(S, gp) # brute-force verifications for all elements of a group S = SymmetricGroup(5) elements = list(S.generate_dimino()) for element in elements: assert _verify_normal_closure(S, element) # small groups small = [] for i in (1, 2, 3): small.append(SymmetricGroup(i)) small.append(AlternatingGroup(i)) small.append(DihedralGroup(i)) small.append(CyclicGroup(i)) for gp in small: for gp2 in small: if gp2.is_subgroup(gp, 0) and gp2.degree == gp.degree: assert _verify_normal_closure(gp, gp2)
def test_stabilizer(): S = SymmetricGroup(2) H = S.stabilizer(0) assert H.generators == [Permutation(1)] a = Permutation([2, 0, 1, 3, 4, 5]) b = Permutation([2, 1, 3, 4, 5, 0]) G = PermutationGroup([a, b]) G0 = G.stabilizer(0) assert G0.order() == 60 gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]] gens = [Permutation(p) for p in gens_cube] G = PermutationGroup(gens) G2 = G.stabilizer(2) assert G2.order() == 6 G2_1 = G2.stabilizer(1) v = list(G2_1.generate(af=True)) assert v == [[0, 1, 2, 3, 4, 5, 6, 7], [3, 1, 2, 0, 7, 5, 6, 4]] gens = ( (1, 2, 0, 4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19), (0, 1, 2, 3, 4, 5, 19, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 17, 18), (0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 16, 11, 12, 13, 14, 15, 8, 17, 10, 19)) gens = [Permutation(p) for p in gens] G = PermutationGroup(gens) G2 = G.stabilizer(2) assert G2.order() == 181440 S = SymmetricGroup(3) assert [G.order() for G in S.basic_stabilizers] == [6, 2]
def test_is_alt_sym(): G = DihedralGroup(10) assert G.is_alt_sym() is False S = SymmetricGroup(10) N_eps = 10 _random_prec = {'N_eps': N_eps, 0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]), 1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]), 2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]), 3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]), 4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]), 5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]), 6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]), 7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]), 8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]), 9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])} assert S.is_alt_sym(_random_prec=_random_prec) is True A = AlternatingGroup(10) _random_prec = {'N_eps': N_eps, 0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]), 1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]), 2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]), 3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]), 4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]), 5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]), 6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]), 7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]), 8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]), 9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])} assert A.is_alt_sym(_random_prec=_random_prec) is False
def test_commutator(): # the commutator of the trivial group and the trivial group is trivial S = SymmetricGroup(3) triv = PermutationGroup([Permutation([0, 1, 2])]) assert S.commutator(triv, triv).is_subgroup(triv) # the commutator of the trivial group and any other group is again trivial A = AlternatingGroup(3) assert S.commutator(triv, A).is_subgroup(triv) # the commutator is commutative for i in (3, 4, 5): S = SymmetricGroup(i) A = AlternatingGroup(i) D = DihedralGroup(i) assert S.commutator(A, D).is_subgroup(S.commutator(D, A)) # the commutator of an abelian group is trivial S = SymmetricGroup(7) A1 = AbelianGroup(2, 5) A2 = AbelianGroup(3, 4) triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])]) assert S.commutator(A1, A1).is_subgroup(triv) assert S.commutator(A2, A2).is_subgroup(triv) # examples calculated by hand S = SymmetricGroup(3) A = AlternatingGroup(3) assert S.commutator(A, S).is_subgroup(A)
def test_pointwise_stabilizer(): S = SymmetricGroup(5) points = [] stab = S for point in (2, 0, 3, 4, 1): stab = stab.stabilizer(point) points.append(point) assert S.pointwise_stabilizer(points).is_subgroup(stab)
def test_verify_bsgs(): S = SymmetricGroup(5) S.schreier_sims() base = S.base strong_gens = S.strong_gens assert _verify_bsgs(S, base, strong_gens) is True assert _verify_bsgs(S, base[:-1], strong_gens) is False assert _verify_bsgs(S, base, S.generators) is False
def test_random_stab(): S = SymmetricGroup(5) _random_el = Permutation([1, 3, 2, 0, 4]) _random_prec = {'rand': _random_el} g = S.random_stab(2, _random_prec=_random_prec) assert g == Permutation([1, 3, 2, 0, 4]) h = S.random_stab(1) assert h(1) == 1
def test_generator_product(): G = SymmetricGroup(5) p = Permutation(0, 2, 3)(1, 4) gens = G.generator_product(p) assert all(g in G.strong_gens for g in gens) w = G.identity for g in gens: w = g*w assert w == p
def test_schreier_vector(): G = CyclicGroup(50) v = [0]*50 v[23] = -1 assert G.schreier_vector(23) == v H = DihedralGroup(8) assert H.schreier_vector(2) == [0, 1, -1, 0, 0, 1, 0, 0] L = SymmetricGroup(4) assert L.schreier_vector(1) == [1, -1, 0, 0]
def test_alt_or_sym(): S = SymmetricGroup(10) A = AlternatingGroup(10) D = DihedralGroup(10) sym = S.alt_or_sym() alt = A.alt_or_sym() dih = D.alt_or_sym() assert sym == 'S' or sym == False assert alt == 'A' or alt == False assert dih == False
def test_pointwise_stabilizer(): S = SymmetricGroup(2) stab = S.pointwise_stabilizer([0]) assert stab.generators == [Permutation(1)] S = SymmetricGroup(5) points = [] stab = S for point in (2, 0, 3, 4, 1): stab = stab.stabilizer(point) points.append(point) assert S.pointwise_stabilizer(points).is_subgroup(stab)
def test_baseswap(): S = SymmetricGroup(4) S.schreier_sims() base = S.base strong_gens = S.strong_gens assert base == [0, 1, 2] deterministic = S.baseswap(base, strong_gens, 1, randomized=False) randomized = S.baseswap(base, strong_gens, 1) assert deterministic[0] == [0, 2, 1] assert _verify_bsgs(S, deterministic[0], deterministic[1]) is True assert randomized[0] == [0, 2, 1] assert _verify_bsgs(S, randomized[0], randomized[1]) is True
def test_remove_gens(): S = SymmetricGroup(10) base, strong_gens = S.schreier_sims_incremental() new_gens = _remove_gens(base, strong_gens) assert _verify_bsgs(S, base, new_gens) is True A = AlternatingGroup(7) base, strong_gens = A.schreier_sims_incremental() new_gens = _remove_gens(base, strong_gens) assert _verify_bsgs(A, base, new_gens) is True D = DihedralGroup(2) base, strong_gens = D.schreier_sims_incremental() new_gens = _remove_gens(base, strong_gens) assert _verify_bsgs(D, base, new_gens) is True
def test_minimal_block(): D = DihedralGroup(6) block_system = D.minimal_block([0, 3]) for i in range(3): assert block_system[i] == block_system[i + 3] S = SymmetricGroup(6) assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0] assert Tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0] P1 = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5)) P2 = PermutationGroup(Permutation(0, 1, 2, 3, 4, 5), Permutation(1, 5)(2, 4)) assert P1.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1] assert P2.minimal_block([0, 2]) == [0, 1, 0, 1, 0, 1]
def test_minimal_block(): D = DihedralGroup(6) block_system = D.minimal_block([0, 3]) for i in range(3): assert block_system[i] == block_system[i + 3] S = SymmetricGroup(6) assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0] assert Tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0] P1 = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5)) P2 = PermutationGroup(Permutation(0, 1, 2, 3, 4, 5), Permutation(1, 5)(2, 4)) assert P1.minimal_block([0, 2]) == [0, 3, 0, 3, 0, 3] assert P2.minimal_block([0, 2]) == [0, 3, 0, 3, 0, 3]
def test_lower_central_series(): # the lower central series of the trivial group consists of the trivial # group triv = PermutationGroup([Permutation([0, 1, 2])]) assert triv.lower_central_series()[0].is_subgroup(triv) # the lower central series of a simple group consists of the group itself for i in (5, 6, 7): A = AlternatingGroup(i) assert A.lower_central_series()[0].is_subgroup(A) # GAP-verified example S = SymmetricGroup(6) series = S.lower_central_series() assert len(series) == 2 assert series[1].is_subgroup(AlternatingGroup(6))
def test_schreier_sims_random(): S = SymmetricGroup(3) base = [0, 1] strong_gens = [Permutation([1, 2, 0]), Permutation([1, 0, 2]),\ Permutation([0, 2, 1])] assert S.schreier_sims_random(base, strong_gens, 5) == (base, strong_gens) D = DihedralGroup(3) _random_prec = {'g': [Permutation([2, 0, 1]), Permutation([1, 2, 0]),\ Permutation([1, 0, 2])]} base = [0, 1] strong_gens = [Permutation([1, 2, 0]), Permutation([2, 1, 0]),\ Permutation([0, 2, 1])] assert D.schreier_sims_random([], D.generators, 2,\ _random_prec=_random_prec) == (base, strong_gens)
def test_derived_series(): # the derived series of the trivial group consists only of the trivial group triv = PermutationGroup([Permutation([0, 1, 2])]) assert triv.derived_series()[0].is_subgroup(triv) # the derived series for a simple group consists only of the group itself for i in (5, 6, 7): A = AlternatingGroup(i) assert A.derived_series()[0].is_subgroup(A) # the derived series for S_4 is S_4 > A_4 > K_4 > triv S = SymmetricGroup(4) series = S.derived_series() assert series[1].is_subgroup(AlternatingGroup(4)) assert series[2].is_subgroup(DihedralGroup(2)) assert series[3].is_trivial
def test_subgroup_search(): prop_true = lambda x: True prop_fix_points = lambda x: [x(point) for point in points] == points prop_comm_g = lambda x: x*g == g*x prop_even = lambda x: x.is_even for i in range(10, 17, 2): S = SymmetricGroup(i) A = AlternatingGroup(i) C = CyclicGroup(i) Sym = S.subgroup_search(prop_true) assert Sym == S Alt = S.subgroup_search(prop_even) assert Alt == A Sym = S.subgroup_search(prop_true, init_subgroup=C) assert Sym == S points = [7] assert S.stabilizer(7) == S.subgroup_search(prop_fix_points) points = [3, 4] assert S.stabilizer(3).stabilizer(4) ==\ S.subgroup_search(prop_fix_points) points = [3, 5] fix35 = A.subgroup_search(prop_fix_points) points = [5] fix5 = A.subgroup_search(prop_fix_points) assert A.subgroup_search(prop_fix_points, init_subgroup=fix35) == fix5 base, strong_gens = A.schreier_sims_incremental() g = A.generators[0] comm_g =\ A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens) assert _verify_bsgs(comm_g, base, comm_g.generators) == True assert [prop_comm_g(gen) == True for gen in comm_g.generators]
def _subgroup_search(i, j, k): prop_true = lambda x: True prop_fix_points = lambda x: [x(point) for point in points] == points prop_comm_g = lambda x: rmul(x, g) == rmul(g, x) prop_even = lambda x: x.is_even for i in range(i, j, k): S = SymmetricGroup(i) A = AlternatingGroup(i) C = CyclicGroup(i) Sym = S.subgroup_search(prop_true) assert Sym.is_subgroup(S) Alt = S.subgroup_search(prop_even) assert Alt.is_subgroup(A) Sym = S.subgroup_search(prop_true, init_subgroup=C) assert Sym.is_subgroup(S) points = [7] assert S.stabilizer(7).is_subgroup(S.subgroup_search(prop_fix_points)) points = [3, 4] assert S.stabilizer(3).stabilizer(4).is_subgroup( S.subgroup_search(prop_fix_points)) points = [3, 5] fix35 = A.subgroup_search(prop_fix_points) points = [5] fix5 = A.subgroup_search(prop_fix_points) assert A.subgroup_search(prop_fix_points, init_subgroup=fix35 ).is_subgroup(fix5) base, strong_gens = A.schreier_sims_incremental() g = A.generators[0] comm_g = \ A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens) assert _verify_bsgs(comm_g, base, comm_g.generators) is True assert [prop_comm_g(gen) is True for gen in comm_g.generators]
def test_is_solvable(): a = Permutation([1, 2, 0]) b = Permutation([1, 0, 2]) G = PermutationGroup([a, b]) assert G.is_solvable G = PermutationGroup([a]) assert G.is_solvable a = Permutation([1, 2, 3, 4, 0]) b = Permutation([1, 0, 2, 3, 4]) G = PermutationGroup([a, b]) assert not G.is_solvable P = SymmetricGroup(10) S = P.sylow_subgroup(3) assert S.is_solvable
def test_sylow_subgroup(): P = PermutationGroup( Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5)) S = P.sylow_subgroup(2) assert S.order() == 4 P = DihedralGroup(12) S = P.sylow_subgroup(3) assert S.order() == 3 P = PermutationGroup( Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5), Permutation(0, 2)) S = P.sylow_subgroup(3) assert S.order() == 9 S = P.sylow_subgroup(2) assert S.order() == 8 P = SymmetricGroup(10) S = P.sylow_subgroup(2) assert S.order() == 256 S = P.sylow_subgroup(3) assert S.order() == 81 S = P.sylow_subgroup(5) assert S.order() == 25 # the length of the lower central series # of a p-Sylow subgroup of Sym(n) grows with # the highest exponent exp of p such # that n >= p**exp exp = 1 length = 0 for i in range(2, 9): P = SymmetricGroup(i) S = P.sylow_subgroup(2) ls = S.lower_central_series() if i // 2**exp > 0: # length increases with exponent assert len(ls) > length length = len(ls) exp += 1 else: assert len(ls) == length G = SymmetricGroup(100) S = G.sylow_subgroup(3) assert G.order() % S.order() == 0 assert G.order() / S.order() % 3 > 0 G = AlternatingGroup(100) S = G.sylow_subgroup(2) assert G.order() % S.order() == 0 assert G.order() / S.order() % 2 > 0
def test_SymmetricGroup(): G = SymmetricGroup(5) elements = list(G.generate()) assert (G.generators[0]).size == 5 assert len(elements) == 120 assert G.is_solvable == False assert G.is_abelian == False assert G.is_transitive == True H = SymmetricGroup(1) assert H.order() == 1 L = SymmetricGroup(2) assert L.order() == 2
def test_verify_centralizer(): # verified by GAP S = SymmetricGroup(3) A = AlternatingGroup(3) triv = PermutationGroup([Permutation([0, 1, 2])]) assert _verify_centralizer(S, S, centr=triv) assert _verify_centralizer(S, A, centr=A)
def test_presentation(): def _test(P): G = P.presentation() return G.order() == P.order() def _strong_test(P): G = P.strong_presentation() chk = len(G.generators) == len(P.strong_gens) return chk and G.order() == P.order() P = PermutationGroup(Permutation(0,1,5,2)(3,7,4,6), Permutation(0,3,5,4)(1,6,2,7)) assert _test(P) P = AlternatingGroup(5) assert _test(P) P = SymmetricGroup(5) assert _test(P) P = PermutationGroup([Permutation(0,3,1,2), Permutation(3)(0,1), Permutation(0,1)(2,3)]) assert _strong_test(P) P = DihedralGroup(6) assert _strong_test(P) a = Permutation(0,1)(2,3) b = Permutation(0,2)(3,1) c = Permutation(4,5) P = PermutationGroup(c, a, b) assert _strong_test(P)
def test_schreier_sims_random(): assert sorted(Tetra.pgroup.base) == [0, 1] S = SymmetricGroup(3) base = [0, 1] strong_gens = [Permutation([1, 2, 0]), Permutation([1, 0, 2]), Permutation([0, 2, 1])] assert S.schreier_sims_random(base, strong_gens, 5) == (base, strong_gens) D = DihedralGroup(3) _random_prec = {'g': [Permutation([2, 0, 1]), Permutation([1, 2, 0]), Permutation([1, 0, 2])]} base = [0, 1] strong_gens = [Permutation([1, 2, 0]), Permutation([2, 1, 0]), Permutation([0, 2, 1])] assert D.schreier_sims_random([], D.generators, 2, _random_prec=_random_prec) == (base, strong_gens)
def test_conjugacy_class(): S = SymmetricGroup(4) x = Permutation(1, 2, 3) C = set( [ Permutation(0, 1, 2, size=4), Permutation(0, 1, 3), Permutation(0, 2, 1, size=4), Permutation(0, 2, 3), Permutation(0, 3, 1), Permutation(0, 3, 2), Permutation(1, 2, 3), Permutation(1, 3, 2), ] ) assert S.conjugacy_class(x) == C
def block_homomorphism(group, blocks): ''' Return the homomorphism induced by the action of the permutation group `group` on the block system `blocks`. The latter should be of the same form as returned by the `minimal_block` method for permutation groups, namely a list of length `group.degree` where the i-th entry is a representative of the block i belongs to. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.named_groups import SymmetricGroup n = len(blocks) # number the blocks; m is the total number, # b is such that b[i] is the number of the block i belongs to, # p is the list of length m such that p[i] is the representative # of the i-th block m = 0 p = [] b = [None]*n for i in range(n): if blocks[i] == i: p.append(i) b[i] = m m += 1 for i in range(n): b[i] = b[blocks[i]] codomain = SymmetricGroup(m) # the list corresponding to the identity permutation in codomain identity = range(m) images = {g: Permutation([b[p[i]^g] for i in identity]) for g in group.generators} H = GroupHomomorphism(group, codomain, images) return H
def test_abelian_invariants(): G = AbelianGroup(2, 3, 4) assert G.abelian_invariants() == [2, 3, 4] G = PermutationGroup( [Permutation(1, 2, 3, 4), Permutation(1, 2), Permutation(5, 6)]) assert G.abelian_invariants() == [2, 2] G = AlternatingGroup(7) assert G.abelian_invariants() == [] G = AlternatingGroup(4) assert G.abelian_invariants() == [3] G = DihedralGroup(4) assert G.abelian_invariants() == [2, 2] G = PermutationGroup([Permutation(1, 2, 3, 4, 5, 6, 7)]) assert G.abelian_invariants() == [7] G = DihedralGroup(12) S = G.sylow_subgroup(3) assert S.abelian_invariants() == [3] G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 3)) assert G.abelian_invariants() == [3] G = PermutationGroup( [Permutation(0, 1), Permutation(0, 2, 4, 6)(1, 3, 5, 7)]) assert G.abelian_invariants() == [2, 4] G = SymmetricGroup(30) S = G.sylow_subgroup(2) assert S.abelian_invariants() == [2, 2, 2, 2, 2, 2, 2, 2, 2, 2] S = G.sylow_subgroup(3) assert S.abelian_invariants() == [3, 3, 3, 3] S = G.sylow_subgroup(5) assert S.abelian_invariants() == [5, 5, 5]
def test_exponent_vector(): Groups = [SymmetricGroup(3), SymmetricGroup(4), SymmetricGroup(9).sylow_subgroup(3), SymmetricGroup(9).sylow_subgroup(2), SymmetricGroup(8).sylow_subgroup(2)] for G in Groups: PcGroup = G.polycyclic_group() collector = PcGroup.collector pcgs = PcGroup.pcgs free_group = collector.free_group for gen in G.generators: exp = collector.exponent_vector(gen) g = Permutation() for i in range(len(exp)): g = g*pcgs[i]**exp[i] if exp[i] else g assert g == gen
def test_minimal_blocks(): P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5)) assert P.minimal_blocks() == [[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]] P = SymmetricGroup(5) assert P.minimal_blocks() == [[0]*5] P = PermutationGroup(Permutation(0, 3)) assert P.minimal_blocks() == False
def test_centralizer(): # the centralizer of the trivial group is the entire group S = SymmetricGroup(2) assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S) A = AlternatingGroup(5) assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A) # a centralizer in the trivial group is the trivial group itself triv = PermutationGroup([Permutation([0, 1, 2, 3])]) D = DihedralGroup(4) assert triv.centralizer(D).is_subgroup(triv) # brute-force verifications for centralizers of groups for i in (4, 5, 6): S = SymmetricGroup(i) A = AlternatingGroup(i) C = CyclicGroup(i) D = DihedralGroup(i) for gp in (S, A, C, D): for gp2 in (S, A, C, D): if not gp2.is_subgroup(gp): assert _verify_centralizer(gp, gp2) # verify the centralizer for all elements of several groups S = SymmetricGroup(5) elements = list(S.generate_dimino()) for element in elements: assert _verify_centralizer(S, element) A = AlternatingGroup(5) elements = list(A.generate_dimino()) for element in elements: assert _verify_centralizer(A, element) D = DihedralGroup(7) elements = list(D.generate_dimino()) for element in elements: assert _verify_centralizer(D, element) # verify centralizers of small groups within small groups small = [] for i in (1, 2, 3): small.append(SymmetricGroup(i)) small.append(AlternatingGroup(i)) small.append(DihedralGroup(i)) small.append(CyclicGroup(i)) for gp in small: for gp2 in small: if gp.degree == gp2.degree: assert _verify_centralizer(gp, gp2)
def test_orbits_transversals_from_bsgs(): S = SymmetricGroup(4) S.schreier_sims() base = S.base strong_gens = S.strong_gens strong_gens_distr = _distribute_gens_by_base(base, strong_gens) result = _orbits_transversals_from_bsgs(base, strong_gens_distr) orbits = result[0] transversals = result[1] base_len = len(base) for i in range(base_len): for el in orbits[i]: assert transversals[i][el](base[i]) == el for j in range(i): assert transversals[i][el](base[j]) == base[j] order = 1 for i in range(base_len): order *= len(orbits[i]) assert S.order() == order
def __init__(self, m, roots): self._m = m self._roots = roots self._value = .0 for i, r in enumerate(self._roots): self._value += r * self._m[i] self._abs = abs(self._value) self._permutation_group = SymmetricGroup(len(roots))
def test_cyclic(): G = SymmetricGroup(2) assert G.is_cyclic G = AbelianGroup(3, 7) assert G.is_cyclic G = AbelianGroup(7, 7) assert not G.is_cyclic G = AlternatingGroup(3) assert G.is_cyclic G = AlternatingGroup(4) assert not G.is_cyclic
def test_composition_series(): a = Permutation(1, 2, 3) b = Permutation(1, 2) G = PermutationGroup([a, b]) comp_series = G.composition_series() assert comp_series == G.derived_series() # The first group in the composition series is always the group itself and # the last group in the series is the trivial group. S = SymmetricGroup(4) assert S.composition_series()[0] == S assert len(S.composition_series()) == 5 A = AlternatingGroup(4) assert A.composition_series()[0] == A assert len(A.composition_series()) == 4 # the composition series for C_8 is C_8 > C_4 > C_2 > triv G = CyclicGroup(8) series = G.composition_series() assert is_isomorphic(series[1], CyclicGroup(4)) assert is_isomorphic(series[2], CyclicGroup(2)) assert series[3].is_trivial
def automorphism_group(graph): """ This is wildly inefficient! :param graph: :return: """ order = graph.order() grp = {Permutation(order)} # The identity group for g in SymmetricGroup(order).generate(): if graph.is_fixed_by(g): grp.add(g) return PermutationGroup(*grp)
def test_presentation(): def _test(P): G = P.presentation() return G.order() == P.order() P = PermutationGroup(Permutation(0,1,5,2)(3,7,4,6), Permutation(0,3,5,4)(1,6,2,7)) assert _test(P) P = AlternatingGroup(5) assert _test(P) P = SymmetricGroup(5) assert _test(P)
def __init__(self, n): self._sym_n = SymmetricGroup(n) self._elements = self._sym_n._elements self._permutation_group = PermutationGroup(self._sym_n._elements) self._path_serialization = os.path.join( os.path.dirname(__file__), 'symmetric_groups/sym_{0}.json'.format(n)) self._subgroups = self.subgroups() self._trivial_group = PermutationGroup( Permutation([i for i in range(0, n)])) with open(self._path_serialization, 'w') as write_file: json.dump(self._subgroups, write_file)
def test_SymmetricGroup(): G = SymmetricGroup(5) elements = list(G.generate()) assert (G.generators[0]).size == 5 assert len(elements) == 120 assert G.is_solvable() == False assert G.is_abelian == False assert G.is_transitive == True H = SymmetricGroup(1) assert H.order() == 1 L = SymmetricGroup(2) assert L.order() == 2
def test_schreier_sims_incremental(): identity = Permutation([0, 1, 2, 3, 4]) TrivialGroup = PermutationGroup([identity]) base, strong_gens = TrivialGroup.schreier_sims_incremental(base=[0, 1, 2]) assert _verify_bsgs(TrivialGroup, base, strong_gens) is True S = SymmetricGroup(5) base, strong_gens = S.schreier_sims_incremental(base=[0, 1, 2]) assert _verify_bsgs(S, base, strong_gens) is True D = DihedralGroup(2) base, strong_gens = D.schreier_sims_incremental(base=[1]) assert _verify_bsgs(D, base, strong_gens) is True A = AlternatingGroup(7) gens = A.generators[:] gen0 = gens[0] gen1 = gens[1] gen1 = rmul(gen1, ~gen0) gen0 = rmul(gen0, gen1) gen1 = rmul(gen0, gen1) base, strong_gens = A.schreier_sims_incremental(base=[0, 1], gens=gens) assert _verify_bsgs(A, base, strong_gens) is True C = CyclicGroup(11) gen = C.generators[0] base, strong_gens = C.schreier_sims_incremental(gens=[gen**3]) assert _verify_bsgs(C, base, strong_gens) is True
def test_cyclic(): G = SymmetricGroup(2) assert G.is_cyclic G = AbelianGroup(3, 7) assert G.is_cyclic G = AbelianGroup(7, 7) assert not G.is_cyclic G = AlternatingGroup(3) assert G.is_cyclic G = AlternatingGroup(4) assert not G.is_cyclic # Order less than 6 G = PermutationGroup(Permutation(0, 1, 2), Permutation(0, 2, 1)) assert G.is_cyclic G = PermutationGroup( Permutation(0, 1, 2, 3), Permutation(0, 2)(1, 3) ) assert G.is_cyclic G = PermutationGroup( Permutation(3), Permutation(0, 1)(2, 3), Permutation(0, 2)(1, 3), Permutation(0, 3)(1, 2) ) assert G.is_cyclic is False # Order 15 G = PermutationGroup( Permutation(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14), Permutation(0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13) ) assert G.is_cyclic # Distinct prime orders assert PermutationGroup._distinct_primes_lemma([3, 5]) is True assert PermutationGroup._distinct_primes_lemma([5, 7]) is True assert PermutationGroup._distinct_primes_lemma([2, 3]) is None assert PermutationGroup._distinct_primes_lemma([3, 5, 7]) is None assert PermutationGroup._distinct_primes_lemma([5, 7, 13]) is True G = PermutationGroup( Permutation(0, 1, 2, 3), Permutation(0, 2)(1, 3)) assert G.is_cyclic assert G._is_abelian
def test_elementary(): a = Permutation([1, 5, 2, 0, 3, 6, 4]) G = PermutationGroup([a]) assert G.is_elementary(7) == False a = Permutation(0, 1)(2, 3) b = Permutation(0, 2)(3, 1) G = PermutationGroup([a, b]) assert G.is_elementary(2) == True c = Permutation(4, 5, 6) G = PermutationGroup([a, b, c]) assert G.is_elementary(2) == False G = SymmetricGroup(4).sylow_subgroup(2) assert G.is_elementary(2) == False H = AlternatingGroup(4).sylow_subgroup(2) assert H.is_elementary(2) == True
def orbit_homomorphism(group, omega): ''' Return the homomorphism induced by the action of the permutation group `group` on the set `omega` that is closed under the action. ''' from sympy.combinatorics import Permutation from sympy.combinatorics.named_groups import SymmetricGroup codomain = SymmetricGroup(len(omega)) identity = codomain.identity omega = list(omega) images = {g: identity*Permutation([omega.index(o^g) for o in omega]) for g in group.generators} group._schreier_sims(base=omega) H = GroupHomomorphism(group, codomain, images) if len(group.basic_stabilizers) > len(omega): H._kernel = group.basic_stabilizers[len(omega)] else: H._kernel = PermutationGroup([group.identity]) return H
def test_sylow_subgroup(): P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5)) S = P.sylow_subgroup(2) assert S.order() == 4 P = DihedralGroup(12) S = P.sylow_subgroup(3) assert S.order() == 3 P = PermutationGroup(Permutation(1, 5)(2, 4), Permutation(0, 1, 2, 3, 4, 5), Permutation(0, 2)) S = P.sylow_subgroup(3) assert S.order() == 9 S = P.sylow_subgroup(2) assert S.order() == 8 P = SymmetricGroup(10) S = P.sylow_subgroup(2) assert S.order() == 256 S = P.sylow_subgroup(3) assert S.order() == 81 S = P.sylow_subgroup(5) assert S.order() == 25 # the length of the lower central series # of a p-Sylow subgroup of Sym(n) grows with # the highest exponent exp of p such # that n >= p**exp exp = 1 length = 0 for i in range(2, 9): P = SymmetricGroup(i) S = P.sylow_subgroup(2) ls = S.lower_central_series() if i // 2**exp > 0: # length increases with exponent assert len(ls) > length length = len(ls) exp += 1 else: assert len(ls) == length G = SymmetricGroup(100) S = G.sylow_subgroup(3) assert G.order() % S.order() == 0 assert G.order()/S.order() % 3 > 0 G = AlternatingGroup(100) S = G.sylow_subgroup(2) assert G.order() % S.order() == 0 assert G.order()/S.order() % 2 > 0
def test_induced_pcgs(): G = SymmetricGroup(9).sylow_subgroup(3) PcGroup = G.polycyclic_group() collector = PcGroup.collector gens = [G[0], G[1]] ipcgs = collector.induced_pcgs(gens) order = [gen.order() for gen in ipcgs] assert order == [3, 3] G = SymmetricGroup(20).sylow_subgroup(2) PcGroup = G.polycyclic_group() collector = PcGroup.collector gens = [G[0], G[1], G[2], G[3]] ipcgs = collector.induced_pcgs(gens) order = [gen.order() for gen in ipcgs] assert order == [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
def test_eq(): a = [[1, 2, 0, 3, 4, 5], [1, 0, 2, 3, 4, 5], [2, 1, 0, 3, 4, 5], [1, 2, 0, 3, 4, 5]] a = [Permutation(p) for p in a + [[1, 2, 3, 4, 5, 0]]] g = Permutation([1, 2, 3, 4, 5, 0]) G1, G2, G3 = [PermutationGroup(x) for x in [a[:2], a[2:4], [g, g ** 2]]] assert G1.order() == G2.order() == G3.order() == 6 assert G1.is_subgroup(G2) assert not G1.is_subgroup(G3) G4 = PermutationGroup([Permutation([0, 1])]) assert not G1.is_subgroup(G4) assert G4.is_subgroup(G1, 0) assert PermutationGroup(g, g).is_subgroup(PermutationGroup(g)) assert SymmetricGroup(3).is_subgroup(SymmetricGroup(4), 0) assert SymmetricGroup(3).is_subgroup(SymmetricGroup(3) * CyclicGroup(5), 0) assert not CyclicGroup(5).is_subgroup(SymmetricGroup(3) * CyclicGroup(5), 0) assert CyclicGroup(3).is_subgroup(SymmetricGroup(3) * CyclicGroup(5), 0)
def test_cmp_perm_lists(): S = SymmetricGroup(4) els = list(S.generate_dimino()) other = els[:] shuffle(other) assert _cmp_perm_lists(els, other) is True
def test_is_primitive(): S = SymmetricGroup(5) assert S.is_primitive() is True C = CyclicGroup(7) assert C.is_primitive() is True