Symmetry Space Group Diagram
You are missing an important part in the quote (the first sentence):


  Fixing the point of inversion at the origin implies that one of the symmetry operators is -x,-y,-z. What are the symmetry operators for the glide plane and screw axis? (The answer is clearly listed, but suppose it was not given here!) The symmetry operator for the c-glide plane is of the form x,T-y,1/2+z, while the symmetry operator for the two-one screw axis is of the form T′-x,1/2+y,T′′-z. It is possible to show (using matrices) that the only values for T, T′, and T′′, which permit the symmetry operators to form a closed group, are 1/2, 0, and 1/2, respectively. This produces the symmetry operators listed on the right-hand side of the space-group diagram shown above. 


Once you fix the origin, T, T' and T'' are indeed fixed. Otherwise, they are not. So to show that they are, you have to combine a glide plane or screw axis with the inversion operation for the chosen origin, and then make an argument.

These arguments usually go as in this example: Two symmetry operations combined give a pure translation. This translation has to be compatible with the crystal lattice, otherwise you have a contradiction.