def test_permutation_methods():
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x * y * x**-1 * y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x * y * x**-1 * y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1 * x * y in S
# Z_5xZ_4
G = FpGroup(F, [x * y * x**-1 * y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x * y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x * y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
