Geodesic
Conjugacy classes of derangements in finite transitive groups
Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 118-123

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Let $G$ be a permutation group acting transitively on a finite set $\Omega $. We classify all such $(G,\Omega )$ when $G$ contains a single conjugacy class of derangements. This was done under the assumption that $G$ acts primitively by Burness and Tong-Viet. It turns out that there are no imprimitive examples. We also discuss some results on the proportion of conjugacy classes which consist of derangements. 
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     author = {Robert M. Guralnick},
     title = {Conjugacy classes of derangements in finite transitive groups},
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     volume = {292},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a6/}
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Robert M. Guralnick. Conjugacy classes of derangements in finite transitive groups. Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 118-123. http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a6/