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_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_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