Transitive permutation groups in which all derangements are involutions. (English) Zbl 1111.20002
A permutation without fixed points is called a ‘derangement’. In the present paper all finite transitive permutation groups are determined in which all derangements are involutions. Also the corresponding problem for abstract groups is considered. There are no surprises. The proof of the main result depends on the classification of finite simple groups.
Reviewer: Wolfgang D. Knapp (Tübingen)
MSC:
| 20B10 | Characterization theorems for permutation groups |
| 20B20 | Multiply transitive finite groups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
Keywords:
finite transitive permutation groups; involutions; derangements; Frobenius groups; elementary Abelian \(2\)-groups; centralizers of involutions; fixed point free permutationsReferences:
| [1] | Aschbacher, M.; Seitz, G. M., Involutions in Chevalley groups over fields of even order, Nagoya Math. J., 63, 1-91 (1976) · Zbl 0359.20014 |
| [2] | Aschbacher, M.; Seitz, G. M., Corrections to ‘Involutions in Chevalley groups over fields of even order’, Nagoya Math. J., 72, 135-136 (1978) · Zbl 0391.20010 |
| [3] | Cameron, P. J.; Cohen, A. M., On the number of fixed point free elements in a permutation group, Discrete Math., 106-107, 135-138 (1992) · Zbl 0813.20001 |
| [4] | Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1984), Oxford University Press: Oxford University Press London · Zbl 0568.20001 |
| [5] | Enomoto, H., The characters of the finite symplectic group \(Sp(4, q), q = 2^f\), Osaka J. Math., 9, 75-94 (1972) · Zbl 0254.20005 |
| [6] | Fein, B.; Kantor, W.; Schacher, M., Relative Brauer groups. II, J. Reine Angew. Math., 328, 39-57 (1981) · Zbl 0457.13004 |
| [7] | Giudici, M., Quasiprimitive groups with no fixed point free elements of prime order, J. London Math. Soc., 67, 73-84 (2003) · Zbl 1050.20002 |
| [8] | Gorenstein, D.; Lyons, R.; Solomon, R., The Classification of the Finite Simple Groups 3 (1998), American Mathematical Society: American Mathematical Society Providence · Zbl 0890.20012 |
| [9] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201 |
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