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The primitive permutation groups of certain degrees. (English) Zbl 0870.20002
The author determines all possibilities for a subgroup of index $$3^rl$$ ($$r\geq 1$$) with $$l\mid 10$$ (that is, $$l=1$$, 2, 5, or 10) in a finite simple group. The classification of finite simple groups is used, together with results of Aschbacher, Liebeck, and others on maximal subgroups of the simple groups. Primitive prime divisors (in factors of the form $$q^n-1$$) are used repeatedly in showing that most maximal subgroups do not have index of the specified form. The author then applies this result, together with the O’Nan-Scott theorem, to classify primitive permutation groups of degree $$3^rl$$ ($$r$$, $$l$$ as above). Finally this result is applied to prove a conjecture of A. Gardiner and C. E. Praeger [J. Algebra 168, No. 3, 798-803 (1994; Zbl 0816.20005)] on transitive groups with “bounded movement” (that conjecture motivated the author’s choice of $$3^rl$$ as the degrees to be classified). This last conjecture was proved independently (in a slightly stronger form) by A. Mann and C. E. Praeger [J. Algebra 181, No. 3, 903-911 (1996; Zbl 0848.20002)], but their proof did not completely classify the possibilities classified in the present paper.

##### MSC:
 20B15 Primitive groups 20B35 Subgroups of symmetric groups 20D05 Finite simple groups and their classification 20E28 Maximal subgroups 20D25 Special subgroups (Frattini, Fitting, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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