Hi,
I think there are different classifications to the general term called the 'genitive'.
These are few things I know about the genitive:
There is the possessive genitive where the notion of possessiveness is manifested. Then, there is a classifying genitive where the part in apostrophes is acting as a modifier or adjective like in 'a girl's school' or 'a learner's manual'.
I think CalifJim mentioned that for possessive cases, the following equation would work well.
the dog's bone -- the bone of the dog
a dog's bone -- the bone of a dog
But this test wouldn't apply for what I call the 'classifying genitive' (should I put quotation marks for the too like this 'the classifying genitive'? How could I know how and when to put quotation marks correctly for the situations like this?), would it?
a Learner's manual -- applying the forementioned test, it would be (if I am right) 'the manual of a Learner' and that wouldn't seem right.
a girl's school -- applying the same test, it would be 'the school of a girl' and that doesn't seem right.
So, only for those cases showing possession are good for the test, right?