×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. math., 76, 469-512, (1984) · Zbl 0537.20023
[2] Aschbacher, M., Overgroups of Sylow subgroups in sporadic groups, Mem. AMS, 60, 343, (1986) · Zbl 0585.20005
[3] Conway, J.H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A., Atlas of finite groups, (1985), Oxford University Press Oxford · Zbl 0568.20001
[4] Cooperstein, B.N., Maximal subgroups of G2(2n), J. algebra, 70, 23-36, (1981) · Zbl 0459.20007
[5] Dixon, J.D.; Mortimer, B., The primitive permutation groups of degree less than 1000, (), 213-238 · Zbl 0646.20003
[6] Gardiner, A.; Praeger, C.E., Transitive permutation groups with bounded movement, J. algebra, 168, 798-803, (1994) · Zbl 0816.20005
[7] Gorenstein, D., Finite simple groups, (1982), Plenum Press New York · Zbl 0182.35402
[8] Guralnick, R.M., Subgroups of prime power index in a simple group, J. algebra, 81, 304-311, (1983) · Zbl 0515.20011
[9] Huppert, B.; Blackburn, N., Finite groups II, (1982), Springer Berlin · Zbl 0477.20001
[10] Kleidman, P.B., ()
[11] Kleidman, P.B., The maximal subgroups of the Chevalley groups G2(q) with q odd, the ree groups 2G2(q) and their automorphism groups, J. algebra, 117, 30-71, (1988) · Zbl 0651.20020
[12] Kleidman, P.B., The maximal subgroups of the finite 8-dimensional orthogonal groups \(PΩ8\^{}\{+\}(q)\) and their automorphism groups, J. algebra, 110, 173-242, (1987) · Zbl 0623.20031
[13] Kleidman, P.B., The maximal subgroups of the Steinberg triality groups ^{3}D4(q) and their automorphism groups, J. algebra, 115, 182-199, (1988) · Zbl 0642.20013
[14] Kleidman, P.B.; Liebeck, M.W., The subgroup structure of the classical groups, () · Zbl 0697.20004
[15] Kleidman, P.B.; Parker, R.A.; Wilson, R.A., The maximal subgroups of the fischer group fi23, J. London math. soc. (2), 39, 89-101, (1989) · Zbl 0629.20006
[16] Kleidman, P.B.; Wilson, R.A., The maximal subgroup of J4, (), 484-510 · Zbl 0619.20004
[17] Liebeck, M.W., On the orders of maximal subgroups of the finite classical groups, (), 426-446 · Zbl 0591.20021
[18] Liebeck, M.W.; Praeger, C.E.; Saxl, J., On the O’nan-Scott theorem for finite primitive permutation groups, J. austral. math. soc. (A), 44, 389-396, (1988) · Zbl 0647.20005
[19] Liebeck, M.W.; Praeger, C.E.; Saxl, J., The maximal factorizations of the finite simple groups and their automorphism groups, Mem. AMS, 86, 432, (1990) · Zbl 0703.20021
[20] Liebeck, M.W.; Saxl, J., The primitive permutation groups of odd degree, J. London math. soc. (2), 31, 250-264, (1985) · Zbl 0573.20004
[21] Liebeck, M.W.; Saxl, J., Primitive permutation groups containing an element of larger prime order, J. London math. soc. (2), 31, 137-149, (1985)
[22] Liebeck, M.W.; Saxl, J., On the orders of maximal subgroups of the finite exceptional groups of Lie type, J. London math. soc. (3), 55, 299-330, (1987) · Zbl 0627.20026
[23] Malle, G., The maximal subgroups of ^{2}F4(q2), J. algebra, 139, 52-69, (1991)
[24] Praeger, C.E., On permutation groups with bounded movement, J. algebra, 144, 436-442, (1991) · Zbl 0744.20004
[25] Shi, W.J., On simple K4-groups, Science sinica, 17, 1281-1283, (1991)
[26] Sims, C.C., Computational methods for permutation groups, (), 169-183
[27] Suzuki, M., Group theory I, (1982), Springer New York
[28] Tan, S.W.; Wang, J., The primitive permutation groups of degree 21 to 30, Beijing daxue xuebao (Chinese), 24, 269-276, (1988) · Zbl 0675.20001
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.