  The power graph of a group is the simple graph with vertices as the group
elements, in which two distinct vertices are adjacent if and only if one of
them can be obtained as an integral power of the other. We study (minimal)
cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which
are associated with its maximal cyclic subgroups. Let $G$ be a finite
non-cyclic nilpotent group whose order is divisible by at least two distinct
primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized
quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, then
under some conditions we prove that there is only one minimum cut-set of the
power graph of $G$. We apply this result to find the vertex connectivity of the
power graphs of certain finite non-cyclic abelian groups whose order is
divisible by at most three distinct primes.
