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_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_cmp_perm_lists(): S = SymmetricGroup(4) els = list(S.generate_dimino()) other = els[:] shuffle(other) assert _cmp_perm_lists(els, other) is True