On the probability that a group element is \(p\)-singular
Let \(G\) be a permutation group of degree \(n\) whose order is divisible by prime \(p\). The authors' Main Theorem states: (a) The probability that an element of \(G\) has order divisible by \(p\) is \(\geq 1/n\). (b) Equality occurs in (a) if and only if \(G\) is either sharply 2-transitive with \(n\) a power of \(p\) or \(S_n\) with \(n = p \geq 5\). It is observed that sufficiency in (b) is easily seen, so that the paper is devoted to the proof of the remainder of the theorem and this begins with a reduction to the case where \(G\) is almost simple and primitive. The proof is completed by investigating the various simple groups, using the classification theorem. The reduction is given in Section 2, the alternating and sporadic groups are treated in Section 3, and Sections 4-10 contain the analysis of the groups of Lie type. (The authors' note that a similar approach was used by \textit{A. Gambini} [Ph. D. thesis, Univ. Freiburg (1992)] but her results were not sharp).
scientific article; zbMATH DE number 802374