Orbital maximin equals size of set for faithful actions by groups with nontrivial center

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Statement

Suppose S is a set of size n. Consider the Orbital maximin problem (?) for faithful group actions by groups with a nontrivial center. In other words, consider the maximum possible value of the size of the smallest orbital under a faithful group action on S by a group with nontrivial center. This maximum equals n. (The maximum is attained by a cyclic group acting on n).

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