+1 to treating the 1/x as a special case in the coercion model.
It would also help with situations like
{{{
sage: F.<x,y>=FreeGroup()
sage: 1/x
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-2-fd72f6c47fe6> in <module>()
----> 1 Integer(1)/x
/home/mmarco/sage/src/sage/structure/element.pyx in
sage.structure.element.RingElement.__div__
(/home/mmarco/sage/src/build/cythonized/sage/structure/element.c:17294)()
1950 if have_same_parent_c(self, right):
1951 return (<RingElement>self)._div_(<RingElement>right)
-> 1952 return coercion_model.bin_op(self, right, div)
1953
1954 cpdef RingElement _div_(self, RingElement right):
/home/mmarco/sage/src/sage/structure/coerce.pyx in
sage.structure.coerce.CoercionModel_cache_maps.bin_op
(/home/mmarco/sage/src/build/cythonized/sage/structure/coerce.c:9915)()
1077 # We should really include the underlying error.
1078 # This causes so much headache.
-> 1079 raise TypeError(arith_error_message(x,y,op))
1080
1081 cpdef canonical_coercion(self, x, y):
TypeError: unsupported operand parent(s) for '/': 'Integer Ring' and 'Free
Group on generators {x, y}'
}}}
You might argue that `1` cannot be coerced into `F`, which is technically
true, but...
{{{
sage: F.one()
1
}}}
So in some sense one can say that `1`is in `F`
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