zbMATH — the first resource for mathematics

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.
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
Full Text: DOI
[1] Boston, N.; Dabrowski, W.; Foguel, T.; Gies, P. J.; Leavitt, J.; Ose, D. T.; Jackson, D. A., The proportion of fixed-point-free elements of a transitive permutation group, Commun. Algebra, 21, 3259-3275, (1993) · Zbl 0783.20002
[2] Burness, T. C.; Tong-Viet, H. P., Derangements in primitive permutation groups, with an application to character theory, Q. J. Math., 66, 63-96, (2015) · Zbl 1335.20001
[3] P. J. Cameron and A. M. Cohen, “On the number of fixed point free elements in a permutation group,” Discrete Math. 106/107, 135-138 (1992). · Zbl 0813.20001
[4] Diaconis, P.; Fulman, J.; Guralnick, R., On fixed points of permutations, J. Algebr. Comb., 28, 189-218, (2008) · Zbl 1192.20001
[5] Erdős, P.; Szalay, M., On some problems of J. denes and P. Turán, 187-212, (1983), Basel · Zbl 0523.10029
[6] Fein, B.; Kantor, W. M.; Schacher, M., Relative Brauer groups. II, J. Reine Angew. Math., 328, 39-57, (1981) · Zbl 0457.13004
[7] Fried, M. D.; Guralnick, R.; Saxl, J., Schur covers and carlitz’s conjecture, Isr. J. Math., 82, 157-225, (1993) · Zbl 0855.11063
[8] Fulman, J.; Guralnick, R., Derangements in simple and primitive groups, 99-121, (2003), River Edge, NJ · Zbl 1036.20002
[9] Fulman, J.; Guralnick, R., Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Am. Math. Soc., 364, 3023-3070, (2012) · Zbl 1256.20048
[10] J. Fulman and R. Guralnick, “Derangements in subspace actions of finite classical groups,” Trans. Am. Math. Soc. (in press); arXiv: 1303.5480 [math.GR]. · Zbl 1431.20033
[11] J. Fulman and R. Guralnick, “Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture,” arXiv: 1508.00039 [math.GR]. · Zbl 06862789
[12] Guralnick, R. M., Zeroes of permutation characters with applications to prime splitting and Brauer groups, J. Algebra, 131, 294-302, (1990) · Zbl 0708.12005
[13] Guralnick, R.; Malle, G., Simple groups admit Beauville structures, J. London Math. Soc., Ser., 85, 694-721, (2012) · Zbl 1255.20009
[14] R. M. Guralnick, P. Müller, and J. Saxl, The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations (Am. Math. Soc., Providence, RI, 2003), Mem. AMS 162 (773). · Zbl 1082.12004
[15] Guralnick, R.; Wan, D., Bounds for fixed point free elements in a transitive group and applications to curves over finite fields, Isr. J. Math., 101, 255-287, (1997) · Zbl 0910.11053
[16] Serre, J.-P., On a theorem of Jordan, Bull. Am. Math. Soc., 40, 429-440, (2003) · Zbl 1047.11045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.