Can a group be a union of three subgroups? I am being asked to show an example of when this fails or prove it rigorously.  I am thinking of using an example to disprove the claim that a group can be a union of three subgroups.  However I am not so sure if proving it rigorously might be a better way.  What do you guys think?  

this may not be much help, but there is a numeric constraint which might serve as a filter (if my understanding of the question  is correct)
if the group is of finite order $g$, the subgroups $g_1,g_2,g_3$, their intersections $g_{12},g_{23},g_{31},g_{123}$ then we require:
$$
g-g_1-g_2-g_3+g_{12}+g_{23}+g_{31} - g_{123} = 0
$$
subject to the constraints of the division lattice implied by Lagrange's theorem.
the quaternion group gives:
$$
8 -4-4-4+2+2+2-2 =0
$$
