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.

中文翻译：

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某些有限非循环群的幂图中的最小割集

群的幂图是以顶点为群元素的简单图，其中两个不同的顶点相邻，当且仅当它们中的一个可以作为另一个的积分幂获得。我们研究（有限）非循环（幂零）群的幂图的（最小）割集，它们与其最大循环子群相关。令 $G$ 是一个有限的非循环幂零群，其阶至少可以被两个不同的素数整除。如果 $G$ 有一个 Sylow 子群，它既不是循环的也不是广义四元数 $2$-group 并且 $G$ 的所有其他 Sylow 子群都是循环的，那么在某些条件下，我们证明只有一个最小的幂割集$G$ 的图表。