def test_isomorphisms(): F, a, b = free_group("a, b") E, c, d = free_group("c, d") # Infinite groups with differently ordered relators. G = FpGroup(F, [a**2, b**3]) H = FpGroup(F, [b**3, a**2]) assert is_isomorphic(G, H) # Trivial Case # FpGroup -> FpGroup H = FpGroup(F, [a**3, b**3, (a*b)**2]) F, c, d = free_group("c, d") G = FpGroup(F, [c**3, d**3, (c*d)**2]) check, T = group_isomorphism(G, H) assert check T(c**3*d**2) == a**3*b**2 # FpGroup -> PermutationGroup # FpGroup is converted to the equivalent isomorphic group. F, a, b = free_group("a, b") G = FpGroup(F, [a**3, b**3, (a*b)**2]) H = AlternatingGroup(4) check, T = group_isomorphism(G, H) assert check assert T(b*a*b**-1*a**-1*b**-1) == Permutation(0, 2, 3) assert T(b*a*b*a**-1*b**-1) == Permutation(0, 3, 2) # PermutationGroup -> PermutationGroup D = DihedralGroup(8) p = Permutation(0, 1, 2, 3, 4, 5, 6, 7) P = PermutationGroup(p) assert not is_isomorphic(D, P) A = CyclicGroup(5) B = CyclicGroup(7) assert not is_isomorphic(A, B) # Two groups of the same prime order are isomorphic to each other. G = FpGroup(F, [a, b**5]) H = CyclicGroup(5) assert G.order() == H.order() assert is_isomorphic(G, H)
def test_isomorphisms(): F, a, b = free_group("a, b") E, c, d = free_group("c, d") # Infinite groups with differently ordered relators. G = FpGroup(F, [a**2, b**3]) H = FpGroup(F, [b**3, a**2]) assert is_isomorphic(G, H) # Trivial Case # FpGroup -> FpGroup H = FpGroup(F, [a**3, b**3, (a*b)**2]) F, c, d = free_group("c, d") G = FpGroup(F, [c**3, d**3, (c*d)**2]) check, T = group_isomorphism(G, H) assert check T(c**3*d**2) == a**3*b**2 # FpGroup -> PermutationGroup # FpGroup is converted to the equivalent isomorphic group. F, a, b = free_group("a, b") G = FpGroup(F, [a**3, b**3, (a*b)**2]) H = AlternatingGroup(4) check, T = group_isomorphism(G, H) assert check assert T(b*a*b**-1*a**-1*b**-1) == Permutation(0, 2, 3) assert T(b*a*b*a**-1*b**-1) == Permutation(0, 3, 2) # PermutationGroup -> PermutationGroup D = DihedralGroup(8) p = Permutation(0, 1, 2, 3, 4, 5, 6, 7) P = PermutationGroup(p) assert not is_isomorphic(D, P) A = CyclicGroup(5) B = CyclicGroup(7) assert not is_isomorphic(A, B) # Two groups of the same prime order are isomorphic to each other. G = FpGroup(F, [a, b**5]) H = CyclicGroup(5) assert G.order() == H.order() assert is_isomorphic(G, H)
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