use the change from #31489

src/sage/groups/abelian_gps/abelian_group_gap.py

@@ -364,11 +364,10 @@ def _element_constructor_(self, x, check=True):
             if len(exp) != len(gens_gap):
                 raise ValueError("input does not match the number of generators")
             x = self.one()
-            for i in range(len(exp)):
-                if orders[i] > 0:
-                    x *= gens_gap[i]**(exp[i] % orders[i])
-                else:
-                    x *= gens_gap[i]**exp[i]
+            for g, e, m in zip(gens_gap, exp, orders):
+                if m != 0:
+                    e = e % m
+                x *= g**e
             x = x.gap()
         return self.element_class(self, x, check=check)
 
@@ -795,8 +794,7 @@ def __init__(self, ambient, gens):
             category = category.Finite()
         else:
             category = category.Infinite()
-        # FIXME: Tell the category that it is a Subobjects() category
-        # category = category.Subobjects()
+        category = category.Subobjects()
         AbelianGroup_gap.__init__(self, G, ambient=ambient, category=category)
 
     def _repr_(self):
@@ -837,6 +835,64 @@ def __reduce__(self):
         gens = tuple([amb(g) for g in self.gens()])
         return amb.subgroup, (gens,)
 
+    def lift(self, x):
+        """
+        Coerce to the ambient group.
+
+        The terminology comes from the category framework and the more general notion of a subquotient.
+
+        INPUT:
+
+        - ``x`` -- an element of this subgroup
+
+        OUTPUT:
+
+        The corresponding element of the ambient group
+
+        EXAMPLES::
+
+            sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
+            sage: G = AbelianGroupGap([4])
+            sage: g = G.gen(0)
+            sage: H = G.subgroup([g^2])
+            sage: h = H.gen(0); h
+            f2
+            sage: h.parent()
+            Subgroup of Abelian group with gap, generator orders (4,) generated by (f2,)
+            sage: H.lift(h)
+            f2
+            sage: H.lift(h).parent()
+            Abelian group with gap, generator orders (4,)
+        """
+        return self.ambient()(x)
+
+    def retract(self, x):
+        """
+        Convert an element of the ambient group into this subgroup.
+
+        The terminology comes from the category framework and the more general notion of a subquotient.
+
+        INPUT:
+
+        - ``x`` -- an element of the ambient group that actually lies in this subgroup.
+
+        OUTPUT:
+
+        The corresponding element of this subgroup
+
+        EXAMPLES::
+
+            sage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap
+            sage: G = AbelianGroupGap([4])
+            sage: g = G.gen(0)
+            sage: H = G.subgroup([g^2])
+            sage: H.retract(g^2)
+            f2
+            sage: H.retract(g^2).parent()
+            Subgroup of Abelian group with gap, generator orders (4,) generated by (f2,)
+        """
+        return self(x)
+
 class AbelianGroupQuotient_gap(AbelianGroup_gap):
     r"""
     Quotients of abelian groups by a subgroup.