Let [Formula: see text] be a finite group and let [Formula: see text] denote the probability that a randomly chosen element from [Formula: see text] fixes a randomly chosen element from [Formula: see text]. We classify all finite abelian groups [Formula: see text] such that [Formula: see text] in the cases when [Formula: see text] is the smallest prime dividing [Formula: see text], and when [Formula: see text] is any prime. We also compute [Formula: see text] for some classes of finite groups. As a consequence of our results, we deduce that if [Formula: see text] is a finite [Formula: see text]-group having a cyclic maximal subgroup, then [Formula: see text] divides [Formula: see text].
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