The Exceptional Hall Numbers

Abstract

A positive integer m is called a Hall number if any finite group of order precisely divisible by m has a Hall subgroup of order m. We prove that, except for the obvious examples, the three integers 12, 24 and 60 are the only Hall numbers, solving a problem proposed by Jiping Zhang.

Keywords

Hall subgroups / Hall numbers / Finite groups / 20D20

References

[1] Bray JN, Holt DF, Roney-Dougal CM. The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, 2013, New York, Cambridge University Press

[2] Conway, J.H., Curtis, R.T., Noton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)

[3] Gorenstein D. The characterization of finite groups with dihedral Sylow 2-subgroups, I.. J. Algebra, 1965, 2(1): 85-151

[4] Gorenstein D. Finite Simple Groups: An Introduction to Their Classification, 1982, New York, Plenum Press

[5] Robinson DJ. A Course in the Theory of Groups, 1982, Berlin, Springer

[6] Serre J-P. A Course in Arithmetic, 1973, Berlin, Springer

Funding

National Natural Science Foundation of China(11931005)

Rights & permissions

Peking University