diff --git a/src/sage/categories/hecke_modules.py b/src/sage/categories/hecke_modules.py
index c21e4e47823..a268e3babe8 100644
--- a/src/sage/categories/hecke_modules.py
+++ b/src/sage/categories/hecke_modules.py
@@ -15,6 +15,7 @@
 from sage.categories.homsets import HomsetsCategory
 from sage.categories.modules_with_basis import ModulesWithBasis
 
+
 class HeckeModules(Category_module):
     r"""
     The category of Hecke modules.
@@ -71,7 +72,7 @@ def __init__(self, R):
         """
         from .commutative_rings import CommutativeRings
         if R not in CommutativeRings():
-            raise TypeError("R (=%s) must be a commutative ring"%R)
+            raise TypeError("R (=%s) must be a commutative ring" % R)
         Category_module.__init__(self, R)
 
     def super_categories(self):
@@ -156,16 +157,10 @@ class Homsets(HomsetsCategory):
         TESTS::
 
             sage: TestSuite(HeckeModules(ZZ).Homsets()).run()
-        """
 
-        def base_ring(self):
-            """
-            EXAMPLES::
-
-                sage: HeckeModules(QQ).Homsets().base_ring()
-                Rational Field
-            """
-            return self.base_category().base_ring()
+            sage: HeckeModules(QQ).Homsets().base_ring()
+            Rational Field
+        """
 
         def extra_super_categories(self):
             """
@@ -180,7 +175,7 @@ def extra_super_categories(self):
                 [Category of vector spaces over Rational Field, Category of homsets]
             """
             from sage.categories.modules import Modules
-            return [Modules(self.base_ring())]
+            return [Modules(self.base_category().base_ring())]
 
         class ParentMethods:
             pass
diff --git a/src/sage/groups/free_group.py b/src/sage/groups/free_group.py
index 6162f225857..9bc5ff4e7d3 100644
--- a/src/sage/groups/free_group.py
+++ b/src/sage/groups/free_group.py
@@ -61,6 +61,7 @@
 # ****************************************************************************
 
 import six
+from sage.categories.groups import Groups
 from sage.groups.group import Group
 from sage.groups.libgap_wrapper import ParentLibGAP, ElementLibGAP
 from sage.structure.unique_representation import UniqueRepresentation
@@ -444,7 +445,7 @@ def fox_derivative(self, gen, im_gens=None, ring=None):
             sage: a.fox_derivative(F([1]),[t,t,t])
             0
         """
-        if not gen in self.parent().generators():
+        if gen not in self.parent().generators():
             raise ValueError("Fox derivative can only be computed with respect to generators of the group")
         l = list(self.Tietze())
         if im_gens is None:
@@ -467,7 +468,7 @@ def fox_derivative(self, gen, im_gens=None, ring=None):
                              # generator of the free group.
         a = R.zero()
         coef = R.one()
-        while len(l) > 0:
+        while l:
             b = l.pop(0)
             if b == i:
                 a += coef * R.one()
@@ -743,6 +744,8 @@ class FreeGroup_class(UniqueRepresentation, Group, ParentLibGAP):
 
         sage: G = FreeGroup('a, b')
         sage: TestSuite(G).run()
+        sage: G.category()
+        Category of infinite groups
     """
     Element = FreeGroupElement
 
@@ -771,7 +774,11 @@ def __init__(self, generator_names, libgap_free_group=None):
         if libgap_free_group is None:
             libgap_free_group = libgap.FreeGroup(generator_names)
         ParentLibGAP.__init__(self, libgap_free_group)
-        Group.__init__(self)
+        if not generator_names:
+            cat = Groups().Finite()
+        else:
+            cat = Groups().Infinite()
+        Group.__init__(self, category=cat)
 
     def _repr_(self):
         """
diff --git a/src/sage/groups/matrix_gps/heisenberg.py b/src/sage/groups/matrix_gps/heisenberg.py
index d113bcb8d2f..4fefcd47ebb 100644
--- a/src/sage/groups/matrix_gps/heisenberg.py
+++ b/src/sage/groups/matrix_gps/heisenberg.py
@@ -116,12 +116,19 @@ def __init__(self, n=1, R=0):
 
             sage: H = groups.matrix.Heisenberg(n=2, R=5)
             sage: TestSuite(H).run()  # long time
+            sage: H.category()
+            Category of finitely generated finite enumerated groups
             sage: H = groups.matrix.Heisenberg(n=2, R=4)
             sage: TestSuite(H).run()  # long time
             sage: H = groups.matrix.Heisenberg(n=3)
             sage: TestSuite(H).run(max_runs=30, skip="_test_elements")  # long time
             sage: H = groups.matrix.Heisenberg(n=2, R=GF(4))
             sage: TestSuite(H).run()  # long time
+
+        TESTS::
+
+            sage: groups.matrix.Heisenberg(n=2, R=ZZ).category()
+            Category of finitely generated infinite enumerated groups
         """
         def elementary_matrix(i, j, val, MS):
             elm = copy(MS.one())
@@ -158,6 +165,8 @@ def elementary_matrix(i, j, val, MS):
         cat = Groups().FinitelyGenerated()
         if self._ring in Rings().Finite():
             cat = cat.Finite()
+        else:
+            cat = cat.Infinite()
 
         FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(dim), self._ring,
                                                   gap_group, category=cat)