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Conjugacy classes of derangements in finite transitive groups. (English) Zbl 1353.20001
Proc. Steklov Inst. Math. 292, 112-117 (2016) and Tr. Mat. Inst. Steklova 292, 118-123 (2016).
Let $$G$$ be a transitive permutation group on a finite set $$\Omega$$ of size at least 2. An element $$g\in G$$ is called a derangement on $$\Omega$$ if it acts fixed-point free on $$\Omega$$. By a theorem of B. Fein et al. [J. Reine Angew. Math. 328, 39–57 (1981; Zbl 0457.13004)], $$G$$ contains a derangement of prime power order. In this paper, the author classifies all such $$(G,\Omega)$$ for which $$G$$ contains a single conjugacy class of derangements. This was done under the assumption that $$G$$ acts primitively on $$\Omega$$ by T. C. Burness and H. P. Tong-Viet [Q. J. Math. 66, No. 1, 63–96 (2015; Zbl 1335.20001)]. It turns out that there are no imprimitive examples. The reduction to the primitive case is an inductive proof and requires the result for almost simple groups. The author also discusses some results on the proportion of conjugacy classes which consist of derangements.
##### MSC:
 20B05 General theory for finite permutation groups 20B20 Multiply transitive finite groups 20D05 Finite simple groups and their classification 20B15 Primitive groups 20C15 Ordinary representations and characters 20E45 Conjugacy classes for groups
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