code: 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
      assert 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)
  
file_path: sympy/sympy/combinatorics/tests/test_homomorphisms.py
imports: from sympy.combinatorics import Permutation
  from sympy.combinatorics.perm_groups import PermutationGroup
  from sympy.combinatorics.homomorphisms import homomorphism, group_isomorphism, is_isomorphic
  from sympy.combinatorics.free_groups import free_group
  from sympy.combinatorics.fp_groups import FpGroup
  from sympy.combinatorics.named_groups import AlternatingGroup, DihedralGroup, CyclicGroup
  from sympy.testing.pytest import raises
label_desc: Use image node_id 2 for calling a global function with example usage: test_isomorphisms() without return types