  We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model
with one hole using the spin-adapted ($SU(2)$ conserving) full configuration
interaction quantum Monte Carlo method. This methodology gives us access to the
ground state energies of all possible spin states $S$ of finite Hubbard
lattices, here obtained for lattices up to 24 sites, for various interaction
strengths ($U$). The critical interaction strength, $U_c$, at which the Nagaoka
transition occurs is determined for each lattice and is found to be
proportional to the lattice size for the larger lattices. Below $U_c$ the
overall ground states are found to favour the minimal total spin ($S=\frac 1
2$), and no intermediate spin state is found to be the overall ground state on
lattices larger than 16 sites. However, at $U_c$, the energies of all the spin
states are found to be nearly degenerate, implying that large fluctuations in
total spin can be expected in the vicinity of the Nagaoka transition.
