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Peter J Cho, Henry H Kim, The Average of the Smallest Prime in a Conjugacy Class, International Mathematics Research Notices, Volume 2020, Issue 6, March 2020, Pages 1718–1747, https://doi.org/10.1093/imrn/rny074
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Abstract
Let C be a conjugacy class of |$S_{n}$| and K an |$S_{n}$|-field. Let |$n_{K,C}$| be the smallest prime, which is ramified or whose Frobenius automorphism Frob|$_{p}$| does not belong to C. Under some technical conjectures, we show that the average of |$n_{K,C}$| is a constant. We explicitly compute the constant. For |$S_{3}$|- and |$S_{4}$|-fields, our result is unconditional. Let |$N_{K,C}$| be the smallest prime for which Frob|$_{p}$| belongs to C. We obtain the average of |$N_{K,C}$| under some technical conjectures. For n = 3 and C = [(12)], we have the average value of |$N_{K,C}$| unconditionally.