## 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
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### 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
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