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Abelian point stabilizers in transitive permutation groups. (English) Zbl 0993.20001

A. Lucchini [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 9, No. 4, 241-243 (1998; Zbl 0940.20006)] proved that if a point-stabilizer \(A\) in a transitive permutation group of degree \(m>1\) is cyclic, then \(|A|<m\). The author proves here a stronger result: if a point-stabilizer \(A\) in a transitive permutation group of degree \(m>1\) is Abelian, then \(\exp(A)<m\).

MSC:

20B20 Multiply transitive finite groups
20B05 General theory for finite permutation groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 0940.20006
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References:

[1] Andrew Chermak and Alberto Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989), no. 4, 907 – 914. · Zbl 0687.20022
[2] Andrea Lucchini, On the order of transitive permutation groups with cyclic point-stabilizer, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 4, 241 – 243 (1999) (English, with English and Italian summaries). · Zbl 0940.20006
[3] V. I. Zenkov, Intersections of abelian subgroups in finite groups, Mat. Zametki 56 (1994), no. 2, 150 – 152 (Russian); English transl., Math. Notes 56 (1994), no. 1-2, 869 – 871 (1995). · Zbl 0839.20032 · doi:10.1007/BF02110750
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