  We derive an exact closed-form expression for fidelity susceptibility of
even- and odd-sized quantum Ising chains in the transverse field. To this aim,
we diagonalize the Ising Hamiltonian and study the gap between its positive and
negative parity subspaces. We derive an exact closed-form expression for the
gap and use it to identify the parity of the ground state. We point out
misunderstanding in some of the former studies of fidelity susceptibility and
discuss its consequences. Last but not least, we rigorously analyze the
properties of the gap. For example, we derive analytical expressions showing
its exponential dependence on the ratio between the system size and the
correlation length.
