×

Minimal permutation representation of the simple group \(F_ 5\). (English. Russian original) Zbl 0659.20015

Algebra Logic 26, No. 3, 167-178 (1987); translation from Algebra Logika 26, No. 3, 298-317 (1987).
Let \(F_ 5\) be the finite simple group of Harada of order \(2^{14}\cdot 3^ 6\cdot 5^ 6\cdot 7\cdot 11\cdot 19\). It contains a subgroup H which is isomorphic to the alternating group \(A_{12}\). The authors prove that H is of minimal possible index in \(F_ 5\) and study the permutation representation of \(F_ 5\) on \(F_ 5/H\). Namely, three theorems are proved:
Theorem 1. The minimal index of a proper subgroup in \(F_ 5\) is \(1,140,000=| F_ 5:H|\). Each subgroup of this index in \(F_ 5\) is conjugate to H. - The permutation representation of \(F_ 5\) on cosets \(F_ 5/H\) has rank 12 (Theorem 2; it also gives subdegrees and stabilizers of two points). Theorem 3. \(F_ 5\neq HA\) for every proper subgroup A in \(F_ 5\). In particular, \(F_ 5\) does not contain wide subgroups.
Reviewer: S.A.Syskin

MSC:

20D08 Simple groups: sporadic groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. D. Mazurov and A. N. Fomin, ”On finite simple nonabelian groups,” Mat. Zametki,34, No. 6, 821–824 (1983). · Zbl 0555.20010
[2] K. Harada, ”On the simple group F of order 214{\(\cdot\)}36{\(\cdot\)}56{\(\cdot\)}7{\(\cdot\)}11{\(\cdot\)}19,” in: Proc. Conf. Finite Groups (Utah, 1975). New York Academic Press (1976), pp. 119–276.
[3] N. P. Mazurova, ”Subgroups of finite groups and the problem of linear optimization,” Algebra Logika,25, No. 4, 405–414 (1986). · Zbl 0631.20008
[4] J. Fischer and J. McKay, ”The nonabelian simple groups G, |G| < 106 – maximal subgroups,” Math. Comp.,32, No. 144, 1293–1302 (1978). · Zbl 0388.20010
[5] J. McKay, ”The non-Abelian simple groups G, |G| < 106 – character tables,” Commun. Algebra,7, No. 13, 1407–1445 (1979). · Zbl 0418.20009
[6] M. Hall, Jr., The Theory of Groups, Macmillan, New York (1959).
[7] B. Huppert, Endliche Gruppen. I, Springer, Berlin (1979).
[8] C. C. Sims, ”Computational methods in the study of permutation groups,” in: Computational Problems in Abstract Algebra (Proc. Conference, Oxford, 1967), Pergamon Press, Oxford (1970), pp. 169–183.
[9] S. P. Norton and R. A. Wilson, ”Maximal subgroups of the Harada-Norton group,” J. Algebra,103, No. 1, 362–376 (1986). · Zbl 0595.20017
[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, An Atlas of Finite Groups, Oxford Univ. Press, London (1985). · Zbl 0568.20001
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.